The Malliavin Calculus and Related Topics

Probability and Its Applications Published in association with the Applied Probability Trust Editors: J. Gani, C.C. He...

Author: David Nualart

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Probability and Its Applications Published in association with the Applied Probability Trust

Editors: J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz

Probability and Its Applications Anderson: Continuous-Time Markov Chains (1991) Azencott/Dacunha-Castelle: Series of Irregular Observations (1986) Bass: Diffusions and Elliptic Operators (1997) Bass: Probabilistic Techniques in Analysis (1995) Chen: Eigenvalues, Inequalities, and Ergodic Theory (2005) Choi: ARMA Model Identiﬁcation (1992) Costa/Fragoso/Marques: Discrete-Time Markov Jump Linear Systems Daley/Vere-Jones: An Introduction of the Theory of Point Processes Volume I: Elementary Theory and Methods, (2nd ed. 2003. Corr. 2nd printing 2005) De la Peña/Giné: Decoupling: From Dependence to Independence (1999) Del Moral: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Durrett: Probability Models for DNA Sequence Evolution (2002) Galambos/Simonelli: Bonferroni-type Inequalities with Applications (1996) Gani (Editor): The Craft of Probabilistic Modelling (1986) Grandell: Aspects of Risk Theory (1991) Gut: Stopped Random Walks (1988) Guyon: Random Fields on a Network (1995) Kallenberg: Foundations of Modern Probability (2nd ed. 2002) Kallenberg: Probabilistic Symmetries and Invariance Principles (2005) Last/Brandt: Marked Point Processes on the Real Line (1995) Leadbetter/Lindgren/Rootzén: Extremes and Related Properties of Random Sequences and Processes (1983) Molchanov: Theory and Random Sets (2005) Nualart: The Malliavin Calculus and Related Topics (2nd ed. 2006) Rachev/Rüschendorf: Mass Transportation Problems Volume I: Theory (1998) Rachev/Rüschendorf: Mass Transportation Problems Volume II: Applications (1998) Resnick: Extreme Values, Regular Variation and Point Processes (1987) Shedler: Regeneration and Networks of Queues (1986) Silvestrov: Limit Theorems for Randomly Stopped Stochastic Processes (2004) Thorisson: Coupling, Stationarity, and Regeneration (2000) Todorovic: An Introduction to Stochastic Processes and Their Applications (1992)

David Nualart

The Malliavin Calculus and Related Topics

ABC

David Nualart Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7523, USA

Series Editors J. Gani

C.C. Heyde

Stochastic Analysis Group, CMA Australian National University Canberra ACT 0200 Australia

Stochastic Analysis Group, CMA Australian National University Canberra ACT 0200 Australia

P. Jagers

T.G. Kurtz

Mathematical Statistics Chalmers University of Technology SE-412 96 Göteborg Sweden

Department of Mathematics University of Wisconsim 480 Lincoln Drive Madison, WI 53706 USA

Library of Congress Control Number: 2005935446

Mathematics Subject Classiﬁcation (2000): 60H07, 60H10, 60H15, 60-02

ISBN-10 3-540-28328-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28328-7 Springer Berlin Heidelberg New York ISBN 0-387-94432-X 1st edition Springer New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and TechBooks using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

SPIN: 11535058

41/TechBooks

543210

To my wife Maria Pilar

Preface to the second edition

There have been ten years since the publication of the ﬁrst edition of this book. Since then, new applications and developments of the Malliavin calculus have appeared. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. The presentation of the Malliavin calculus has been slightly modiﬁed at some points, where we have taken advantage of the material from the lectures given in Saint Flour in 1995 (see reference [248]). The main changes and additional material are the following: In Chapter 1, the derivative and divergence operators are introduced in the framework of an isonormal Gaussian process associated with a general Hilbert space H. The case where H is an L2 -space is trated in detail afterwards (white noise case). The Sobolev spaces Ds,p , with s is an arbitrary real number, are introduced following Watanabe’s work. Chapter 2 includes a general estimate for the density of a one-dimensional random variable, with application to stochastic integrals. Also, the composition of tempered distributions with nondegenerate random vectors is discussed following Watanabe’s ideas. This provides an alternative proof of the smoothness of densities for nondegenerate random vectors. Some properties of the support of the law are also presented. In Chapter 3, following the work by Al` os and Nualart [10], we have included some recent developments on the Skorohod integral and the associated change-of-variables formula for processes with are diﬀerentiable in future times. Also, the section on substitution formulas has been rewritten

viii

Preface to the second edition

and an Ito-Ventzell ˆ formula has been added, following [248]. This formula allows us to solve anticipating stochastic diﬀerential equations in Stratonovich sense with random initial condition. There have been only minor changes in Chapter 4, and two additional chapters have been included. Chapter 5 deals with the stochastic calculus with respect to the fractional Brownian motion. The fractional Brownian motion is a self-similar Gaussian process with stationary increments and variance t2H . The parameter H ∈ (0, 1) is called the Hurst parameter. The main purpose of this chapter is to use the the Malliavin Calculus techniques to develop a stochastic calculus with respect to the fractional Brownian motion. Finally, Chapter 6 contains some applications of Malliavin Calculus in Mathematical Finance. The integration-by-parts formula is used to compute “greeks”, sensitivity parameters of the option price with respect to the underlying parameters of the model. We also discuss the application of the Clark-Ocone formula in hedging derivatives and the additional expected logarithmic utility for insider traders.

August 20, 2005

David Nualart

Preface

The origin of this book lies in an invitation to give a series of lectures on Malliavin calculus at the Probability Seminar of Venezuela, in April 1985. The contents of these lectures were published in Spanish in [245]. Later these notes were completed and improved in two courses on Malliavin cal´ culus given at the University of California at Irvine in 1986 and at Ecole Polytechnique F´ ´ed´erale de Lausanne in 1989. The contents of these courses correspond to the material presented in Chapters 1 and 2 of this book. Chapter 3 deals with the anticipating stochastic calculus and it was developed from our collaboration with Moshe Zakai and Etienne Pardoux. The series of lectures given at the Eighth Chilean Winter School in Probability and Statistics, at Santiago de Chile, in July 1989, allowed us to write a pedagogical approach to the anticipating calculus which is the basis of Chapter 3. Chapter 4 deals with the nonlinear transformations of the Wiener measure and their applications to the study of the Markov property for solutions to stochastic diﬀerential equations with boundary conditions. The presentation of this chapter was inspired by the lectures given at the Fourth Workshop on Stochastic Analysis in Oslo, in July 1992. I take the opportunity to thank these institutions for their hospitality, and in particular I would like to thank Enrique Caba˜ n ˜a, Mario Wschebor, Joaqu´n ¨ ¨ Ortega, S¨ u ¨leyman Ustunel, Bernt Øksendal, Renzo Cairoli, Ren´ ´e Carmona, and Rolando Rebolledo for their invitations to lecture on these topics. We assume that the reader has some familiarity with the Itˆo stochastic calculus and martingale theory. In Section 1.1.3 an introduction to the Itˆ o calculus is provided, but we suggest the reader complete this outline of the classical Itˆo calculus with a review of any of the excellent presentations of

x

Preface

this theory that are available (for instance, the books by Revuz and Yor [292] and Karatzas and Shreve [164]). In the presentation of the stochastic calculus of variations (usually called the Malliavin calculus) we have chosen the framework of an arbitrary centered Gaussian family, and have tried to focus our attention on the notions and results that depend only on the covariance operator (or the associated Hilbert space). We have followed some of the ideas and notations developed by Watanabe in [343] for the case of an abstract Wiener space. In addition to Watanabe’s book and the survey on the stochastic calculus of variations written by Ikeda and Watanabe in [144] we would like to mention the book by Denis Bell [22] (which contains a survey of the diﬀerent approaches to the Malliavin calculus), and the lecture notes by Dan Ocone in [270]. Readers interested in the Malliavin calculus for jump processes can consult the book by Bichteler, Gravereaux, and Jacod [35]. The objective of this book is to introduce the reader to the Sobolev differential calculus for functionals of a Gaussian process. This is called the analysis on the Wiener space, and is developed in Chapter 1. The other chapters are devoted to diﬀerent applications of this theory to problems such as the smoothness of probability laws (Chapter 2), the anticipating stochastic calculus (Chapter 3), and the shifts of the underlying Gaussian process (Chapter 4). Chapter 1, together with selected parts of the subsequent chapters, might constitute the basis for a graduate course on this subject. I would like to express my gratitude to the people who have read the several versions of the manuscript, and who have encouraged me to complete the work, particularly I would like to thank John Walsh, Giuseppe Da Prato, Moshe Zakai, and Peter Imkeller. My special thanks go to Michael Rockner ¨ for his careful reading of the ﬁrst two chapters of the manuscript. March 17, 1995

David Nualart

Contents

Introduction 1 Analysis on the Wiener space 1.1 Wiener chaos and stochastic integrals . . . . . . . . . . . . 1.1.1 The Wiener chaos decomposition . . . . . . . . . . . 1.1.2 The white noise case: Multiple Wiener-Itˆo integrals . 1.1.3 Itˆ oˆ stochastic calculus . . . . . . . . . . . . . . . . . 1.2 The derivative operator . . . . . . . . . . . . . . . . . . . . 1.2.1 The derivative operator in the white noise case . . . 1.3 The divergence operator . . . . . . . . . . . . . . . . . . . . 1.3.1 Properties of the divergence operator . . . . . . . . . 1.3.2 The Skorohod integral . . . . . . . . . . . . . . . . . 1.3.3 The Itoˆ stochastic integral as a particular case of the Skorohod integral . . . . . . . . . . . . . . . . 1.3.4 Stochastic integral representation of Wiener functionals . . . . . . . . . . . . . . . . . 1.3.5 Local properties . . . . . . . . . . . . . . . . . . . . 1.4 The Ornstein-Uhlenbeck semigroup . . . . . . . . . . . . . . 1.4.1 The semigroup of Ornstein-Uhlenbeck . . . . . . . . 1.4.2 The generator of the Ornstein-Uhlenbeck semigroup 1.4.3 Hypercontractivity property and the multiplier theorem . . . . . . . . . . . . . . 1.5 Sobolev spaces and the equivalence of norms . . . . . . . .

1 3 3 4 8 15 24 31 36 37 40 44 46 47 54 54 58 61 67

xii

Contents

2 Regularity of probability laws 2.1 Regularity of densities and related topics . . . . . . . . . . . 2.1.1 Computation and estimation of probability densities 2.1.2 A criterion for absolute continuity based on the integration-by-parts formula . . . . . . 2.1.3 Absolute continuity using Bouleau and Hirsch’s approach . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Smoothness of densities . . . . . . . . . . . . . . . . 2.1.5 Composition of tempered distributions with nondegenerate random vectors . . . . . . . . . . . . . . . . 2.1.6 Properties of the support of the law . . . . . . . . . 2.1.7 Regularity of the law of the maximum of continuous processes . . . . . . . . . . . . . . . . 2.2 Stochastic diﬀerential equations . . . . . . . . . . . . . . . . 2.2.1 Existence and uniqueness of solutions . . . . . . . . 2.2.2 Weak diﬀerentiability of the solution . . . . . . . . . 2.3 Hypoellipticity and H¨ o¨rmander’s theorem . . . . . . . . . . 2.3.1 Absolute continuity in the case of Lipschitz coeﬃcients . . . . . . . . . . . . . . . . 2.3.2 Absolute continuity under H¨ ormander’s conditions . 2.3.3 Smoothness of the density under H¨ o¨rmander’s condition . . . . . . . . . . . . . 2.4 Stochastic partial diﬀerential equations . . . . . . . . . . . . 2.4.1 Stochastic integral equations on the plane . . . . . . 2.4.2 Absolute continuity for solutions to the stochastic heat equation . . . . . . . . . . . . 3 Anticipating stochastic calculus 3.1 Approximation of stochastic integrals . . . . . . . . . . . . . 3.1.1 Stochastic integrals deﬁned by Riemann sums . . . . 3.1.2 The approach based on the L2 development of the process . . . . . . . . . . . . . . . . . . . . . . 3.2 Stochastic calculus for anticipating integrals . . . . . . . . . 3.2.1 Skorohod integral processes . . . . . . . . . . . . . . 3.2.2 Continuity and quadratic variation of the Skorohod integral . . . . . . . . . . . . . . . . 3.2.3 Itˆ oˆ’s formula for the Skorohod and Stratonovich integrals . . . . . . . . . . . . . . . 3.2.4 Substitution formulas . . . . . . . . . . . . . . . . . 3.3 Anticipating stochastic diﬀerential equations . . . . . . . . 3.3.1 Stochastic diﬀerential equations in the Sratonovich sense . . . . . . . . . . . . . . . . 3.3.2 Stochastic diﬀerential equations with boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 90 94 99 104 105 108 116 117 119 125 125 128 133 142 142 151 169 169 170 176 180 180 181 184 195 208 208 215

Contents

3.3.3

xiii

Stochastic diﬀerential equations in the Skorohod sense . . . . . . . . . . . . . . . . . 217

4 Transformations of the Wiener measure 4.1 Anticipating Girsanov theorems . . . . . . . . . . . . . . 4.1.1 The adapted case . . . . . . . . . . . . . . . . . . 4.1.2 General results on absolute continuity of transformations . . . . . . . . . . . . . . . . . 4.1.3 Continuously diﬀerentiable variables in the direction of H 1 . . . . . . . . . . . . . . . 4.1.4 Transformations induced by elementary processes 4.1.5 Anticipating Girsanov theorems . . . . . . . . . . 4.2 Markov random ﬁelds . . . . . . . . . . . . . . . . . . . 4.2.1 Markov ﬁeld property for stochastic diﬀerential equations with boundary conditions . . . . . . . 4.2.2 Markov ﬁeld property for solutions to stochastic partial diﬀerential equations . . . . 4.2.3 Conditional independence and factorization properties . . . . . . . . . . . .

225 . . 225 . . 226 . . 228 . . . .

. . . .

230 232 234 241

. . 242 . . 249 . . 258

5 Fractional Brownian motion 273 5.1 Deﬁnition, properties and construction of the fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.1.1 Semimartingale property . . . . . . . . . . . . . . . . 274 5.1.2 Moving average representation . . . . . . . . . . . . 276 5.1.3 Representation of fBm on an interval . . . . . . . . . 277 5.2 Stochastic calculus with respect to fBm . . . . . . . . . . . 287 5.2.1 Malliavin Calculus with respect to the fBm . . . . . 287 5.2.2 Stochastic calculus with respect to fBm. Case H > 12 288 5.2.3 Stochastic integration with respect to fBm in the case H < 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.3 Stochastic diﬀerential equations driven by a fBm . . . . . . 306 5.3.1 Generalized Stieltjes integrals . . . . . . . . . . . . . 306 5.3.2 Deterministic diﬀerential equations . . . . . . . . . . 309 5.3.3 Stochastic diﬀerential equations with respect to fBm 312 5.4 Vortex ﬁlaments based on fBm . . . . . . . . . . . . . . . . 313 6 Malliavin Calculus in ﬁnance 6.1 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . 6.1.1 Arbitrage opportunities and martingale measures . 6.1.2 Completeness and hedging . . . . . . . . . . . . . . 6.1.3 Black-Scholes formula . . . . . . . . . . . . . . . . 6.2 Integration by parts formulas and computation of Greeks 6.2.1 Computation of Greeks for European options . . . 6.2.2 Computation of Greeks for exotic options . . . . .

. . . . . . .

321 321 323 325 327 330 332 334

xiv

Contents

6.3

6.4

Application of the Clark-Ocone formula in 6.3.1 A generalized Clark-Ocone formula 6.3.2 Application to ﬁnance . . . . . . . Insider trading . . . . . . . . . . . . . . .

A Appendix A.1 A Gaussian formula . . . . . . . . A.2 Martingale inequalities . . . . . . . A.3 Continuity criteria . . . . . . . . . A.4 Carleman-Fredholm determinant . A.5 Fractional integrals and derivatives

. . . . .

. . . . .

. . . . .

. . . . .

hedging . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

336 336 338 340

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

351 351 351 353 354 355

. . . . .

. . . . .

. . . . .

. . . . .

References

357

Index

377

Introduction

The Malliavin calculus (also known as the stochastic calculus of variations) is an inﬁnite-dimensional diﬀerential calculus on the Wiener space. It is tailored to investigate regularity properties of the law of Wiener functionals such as solutions of stochastic diﬀerential equations. This theory was initiated by Malliavin and further developed by Stroock, Bismut, Watanabe, and others. The original motivation, and the most important application of this theory, has been to provide a probabilistic proof of H¨ ormander’s “sum of squares” theorem. One can distinguish two parts in the Malliavin calculus. First is the theory of the diﬀerential operators deﬁned on suitable Sobolev spaces of Wiener functionals. A crucial fact in this theory is the integration-by-parts formula, which relates the derivative operator on the Wiener space and the Skorohod extended stochastic integral. A second part of this theory deals with establishing general criteria in terms of the “Malliavin covariance matrix” for a given random vector to possess a density or, even more precisely, a smooth density. In the applications of Malliavin calculus to speciﬁc examples, one usually tries to ﬁnd suﬃcient conditions for these general criteria to be fulﬁlled. In addition to the study of the regularity of probability laws, other applications of the stochastic calculus of variations have recently emerged. For instance, the fact that the adjoint of the derivative operator coincides with a noncausal extension of the Itˆo stochastic integral introduced by Skorohod is the starting point in developing a stochastic calculus for nonadapted processes, which is similar in some aspects to the Itˆˆo calculus. This anticipating stochastic calculus has allowed mathematicians to formulate and

2

Introduction

discuss stochastic diﬀerential equations where the solution is not adapted to the Brownian ﬁltration. The purposes of this monograph are to present the main features of the Malliavin calculus, including its application to the proof of H¨ o¨rmander’s theorem, and to discuss in detail its connection with the anticipating stochastic calculus. The material is organized in the following manner: In Chapter 1 we develop the analysis on the Wiener space (Malliavin calculus). The ﬁrst section presents the Wiener chaos decomposition. In Sections 2,3, and 4 we study the basic operators D, δ, and L, respectively. The operator D is the derivative operator, δ is the adjoint of D, and L is the generator of the Ornstein-Uhlenbeck semigroup. The last section of this chapter is devoted to proving Meyer’s equivalence of norms, following a simple approach due to Pisier. We have chosen the general framework of an isonormal Gaussian process {W (h), h ∈ H} associated with a Hilbert space H. The particular case where H is an L2 space over a measure space (T, B, µ) (white noise case) is discussed in detail. Chapter 2 deals with the regularity of probability laws by means of the Malliavin calculus. In Section 3 we prove H¨ o¨rmander’s theorem, using the general criteria established in the ﬁrst sections. Finally, in the last section we discuss the regularity of the probability law of the solutions to hyperbolic and parabolic stochastic partial diﬀerential equations driven by a spacetime white noise. In Chapter 3 we present the basic elements of the stochastic calculus for anticipating processes, and its application to the solution of anticipating stochastic diﬀerential equations. Chapter 4 examines diﬀerent extensions of the Girsanov theorem for nonlinear and anticipating transformations of the Wiener measure, and their application to the study of the Markov property of solution to stochastic diﬀerential equations with boundary conditions. Chapter 5 deals with some recent applications of the Malliavin Calculus to develop a stochastic calculus with respect to the fractional Brownian motion. Finally, Chapter 6 presents some applications of the Malliavin Calculus in Mathematical Finance. The appendix contains some basic results such as martingale inequalities and continuity criteria for stochastic processes that are used along the book.

1 Analysis on the Wiener space

In this chapter we study the diﬀerential calculus on a Gaussian space. That is, we introduce the derivative operator and the associated Sobolev spaces of weakly diﬀerentiable random variables. Then we prove the equivalence of norms established by Meyer and discuss the relationship between the basic diﬀerential operators: the derivative operator, its adjoint (which is usually called the Skorohod integral), and the Ornstein-Uhlenbeck operator.

1.1 Wiener chaos and stochastic integrals This section describes the basic framework that will be used in this monograph. The general context consists of a probability space (Ω, F, P ) and a Gaussian subspace H1 of L2 (Ω, F, P ). That is, H1 is a closed subspace whose elements are zero-mean Gaussian random variables. Often it will be convenient to assume that H1 is isometric to an L2 space of the form L2 (T, B, µ), where µ is a σ-ﬁnite measure without atoms. In this way the elements of H1 can be interpreted as stochastic integrals of functions in L2 (T, B, µ) with respect to a random Gaussian measure on the parameter space T (Gaussian white noise). In the ﬁrst part of this section we obtain the orthogonal decomposition into the Wiener chaos for square integrable functionals of our Gaussian process. The second part is devoted to the construction and main properties of multiple stochastic integrals with respect to a Gaussian white noise. Finally, in the third part we recall some basic facts about the Itˆ o integral.

4

1. Analysis on the Wiener space

1.1.1 The Wiener chaos decomposition Suppose that H is a real separable Hilbert space with scalar product denoted by ·, ·H . The norm of an element h ∈ H will be denoted by hH . Deﬁnition 1.1.1 We say that a stochastic process W = {W (h), h ∈ H} deﬁned in a complete probability space (Ω, F, P ) is an isonormal Gaussian process (or a Gaussian process on H) if W is a centered Gaussian family of random variables such that E(W (h)W (g)) = h, gH for all h, g ∈ H. Remarks: 1. Under the above conditions, the mapping h → W (h) is linear. Indeed, for any λ, µ ∈ R, and h, g ∈ H, we have E (W (λh + µg) − λW (h) − µW (g))2 = λh + µg2H +λ2 h2H + µ2 g2H − 2λλh + µg, hH −2µλh + µg, gH + 2λµh, gH = 0. The mapping h → W (h) provides a linear isometry of H onto a closed subspace of L2 (Ω, F, P ) that we will denote by H1 . The elements of H1 are zero-mean Gaussian random variables. 2. In Deﬁnition 1.1.1 it is enough to assume that each random variable W (h) is Gaussian and centered, since by Remark 1 the mapping h → W (h) is linear, which implies that {W (h)} is a Gaussian family. 3. By Kolmogorov’s theorem, given the Hilbert space H we can always construct a probability space and a Gaussian process {W (h)} verifying the above conditions. Let Hn (x) denote the nth Hermite polynomial, which is deﬁned by Hn (x) =

(−1)n x2 dn − x2 e2 (e 2 ), n! dxn

n ≥ 1,

and H0 (x) = 1. These polynomials are the coeﬃcients of the expansion in 2 powers of t of the function F (x, t) = exp(tx − t2 ). In fact, we have F (x, t)

=

= e =

1 x2 − (x − t)2 ] 2 2 ∞ n t dn − (x−t)2 2 ( e )|t=0 n! dtn n=0

exp[ x2 2

∞ n=0

tn Hn (x).

(1.1)

1.1 Wiener chaos and stochastic integrals

5

Using this development, one can easily show the following properties: Hn (x) = Hn−1 (x),

n ≥ 1,

(n + 1)H Hn+1 (x) = xH Hn (x) − Hn−1 (x), Hn (−x) = (−1)n Hn (x),

(1.2) n ≥ 1,

n ≥ 1.

(1.3) (1.4)

∂F Indeed, (1.2) and (1.3) follow from ∂F ∂x = tF , respectively, and ∂t = (x − t)F , and (1.4) is a consequence of F (−x, t) = F (x, −t). The ﬁrst Hermite polynomials are H1 (x) = x and H2 (x) = 12 (x2 − 1). n From (1.3) it follows that the highest-order term of Hn (x) is xn! . Also, from 2 the expansion of F (0, t) = exp(− t2 ) in powers of t, we get Hn (0) = 0 if n k

is odd and H2k (0) = (−1) for all k ≥ 1. The relationship between Hermite 2k k! polynomials and Gaussian random variables is explained by the following result. Lemma 1.1.1 Let X, Y be two random variables with joint Gaussian distribution such that E(X) = E(Y ) = 0 and E(X 2 ) = E(Y 2 ) = 1. Then for all n, m ≥ 0 we have 0 if n = m. E(H Hn (X)H Hm (Y )) = 1 n (E(XY )) if n = m. n! Proof:

For all s, t ∈ R we have s2 t2 E exp(sX − ) exp(tY − ) = exp(stE(XY )). 2 2 n+m

∂ Taking the (n + m)th partial derivative ∂s n ∂tm at s = t = 0 in both sides of the above equality yields 0 if n = m. E(n!m!H Hn (X)H Hm (Y )) = n!(E(XY ))n if n = m.

We will denote by G the σ-ﬁeld generated by the random variables {W (h), h ∈ H}. Lemma 1.1.2 The random variables {eW (h) , h ∈ H} form a total subset of L2 (Ω, G, P ). Proof: Let X ∈ L2 (Ω, G, P ) be such that E(XeW (h) ) = 0 for all h ∈ H. The linearity of the mapping h → W (h) implies m ti W (hi ) = 0 (1.5) E X exp i=1

6

1. Analysis on the Wiener space

for any t1 , . . . , tm ∈ R, h1 , . . . , hm ∈ H, m ≥ 1. Suppose that m ≥ 1 and h1 , . . . , hm ∈ H are ﬁxed. Then Eq. (1.5) says that the Laplace transform of the signed measure ν(B) = E (X1B (W (h1 ), . . . , W (hm ))) , where B is a Borel subset of Rm , is identically zero on Rm . Consequently, this measure is zero, which implies E(X1G ) = 0 for any G ∈ G. So X = 0, completing the proof of the lemma. For each n ≥ 1 we will denote by Hn the closed linear subspace of Hn (W (h)), h ∈ H, hH = L2 (Ω, F, P ) generated by the random variables {H 1}. H0 will be the set of constants. For n = 1, H1 coincides with the set of random variables {W (h), h ∈ H}. From Lemma 1.1.1 we deduce that the subspaces Hn and Hm are orthogonal whenever n = m. The space Hn is called the Wiener chaos of order n, and we have the following orthogonal decomposition. Theorem 1.1.1 Then the space L2 (Ω, G, P ) can be decomposed into the inﬁnite orthogonal sum of the subspaces Hn : L2 (Ω, G, P ) = ⊕∞ n=0 Hn . Proof: Let X ∈ L2 (Ω, G, P ) such that X is orthogonal to Hn for all n ≥ 0. We want to show that X = 0. We have E(XH Hn (W (h))) = 0 for all h ∈ H with hH = 1. Using the fact that xn can be expressed as a linear combination of the Hermite polynomials Hr (x), 0 ≤ r ≤ n, we get E(XW (h)n ) = 0 for all n ≥ 0, and therefore E(X exp(tW (h))) = 0 for all t ∈ R, and for all h ∈ H of norm one. By Lemma 1.1.2 we deduce X = 0, which completes the proof of the theorem. 0 For any n ≥ 1 we can consider the space Pn formed by the random variables p(W (h1 ), . . . , W (hk )), where k ≥ 1, h1 , . . . , hk ∈ H, and p is a real polynomial in k variables of degree less than or equal to n. Let Pn be the closure of Pn0 in L2 . Then it holds that H0 ⊕H1 ⊕· · ·⊕Hn = Pn . In fact, the inclusion ⊕ni=0 Hi ⊂ Pn is immediate. To prove the converse inclusion, it suﬃces to check that Pn is orthogonal to Hm for all m > n. We want to Hm (W (h))) = 0, where hH = 1, p is show that E(p(W (h1 ), . . . , W (hk ))H a polynomial of degree less than or equal to n, and m > n. We can replace p(W (h1 ), . . . , W (hk )) by q(W (e1 ), . . . , W (ej ), W (h)), where {e1 , . . . , ej , h} is an orthonormal family and the degree of q is less than or equal to n. Then it remains to show only that E(W (h)r Hm (W (h))) = 0 for all r ≤ n < m; this is immediate because xr can be expressed as a linear combination of the Hermite polynomials Hq (x), 0 ≤ q ≤ r. We denote by Jn the projection on the nth Wiener chaos Hn . Example 1.1.1 Consider the following simple example, which corresponds to the case where the Hilbert space H is one-dimensional. Let (Ω, F, P ) =

1.1 Wiener chaos and stochastic integrals

7

(R, B(R), ν), where ν is the standard normal law N (0, 1). Take H = R, and for any h ∈ R set W (h)(x) = hx. There are only two elements in H of norm one: 1 and −1. We associate with them the random variables x and −x, respectively. From (1.4) it follows that Hn has dimension one and is generated by Hn (x). In this context, Theorem 1.1.1 means that the Hermite polynomials form a complete orthonormal system in L2 (R, ν). Suppose now that H is inﬁnite-dimensional (the ﬁnite-dimensional case would be similar and easier), and let {ei , i ≥ 1} be an orthonormal basis of H. We will denote by Λ the set of all sequences a = (a1 , a2 , . . . ), ai ∈ N, such that all the terms, except

a ﬁnite number of them, vanish. For a ∈ Λ ∞ ∞ we set a! = i=1 ai ! and |a| = i=1 ai . For any multiindex a ∈ Λ we deﬁne the generalized Hermite polynomial Ha (x), x ∈ RN , by Ha (x) =

∞

Hai (xi ).

i=1

The above product is well deﬁned because H0 (x) = 1 and ai = 0 only for a ﬁnite number of indices. For any a ∈ Λ we deﬁne Φa =

∞ √ a! Hai (W (ei )).

(1.6)

i=1

The family of random variables {Φa , a ∈ Λ} is an orthonormal system. Indeed, for any a, b ∈ Λ we have

E

∞

Hai (W (ei ))H Hbi (W (ei ))

=

i=1

∞

E(H Hai (W (ei ))H Hbi (W (ei )))

i=1

=

1 a!

0

if if

a = b. a = b.

(1.7)

Proposition 1.1.1 For any n ≥ 1 the random variables {Φa , a ∈ Λ, |a| = n}

(1.8)

form a complete orthonormal system in Hn . Proof: Observe that when n varies, the families (1.8) are mutually orthogonal in view of (1.7). On the other hand, the random variables of the family (1.8) belong to Pn . Then it is enough to show that every polynomial random variable p(W (h1 ), . . . , W (hk )) can be approximated by polynomials in W (ei ), which is clear because {ei , i ≥ 1} is a basis of H. As a consequence of Proposition 1.1.1 the family {Φa , a ∈ Λ} is a complete orthonormal system in L2 (Ω, G, P ).

8

1. Analysis on the Wiener space

Let a ∈ Λ be a multiindex such that |a| = n. The mapping √ ⊗ai = a!Φa In symm ⊗∞ i=1 ei

(1.9)

provides an isometry between the symmetric tensor product H ⊗n , equipped √ with the norm n! ·H ⊗n , and the nth Wiener chaos Hn . In fact, 2

a! n! a! ⊗ai 2 ⊗ai 2

symm ⊗∞

⊗∞ = e = ⊗n i=1 ei i=1 i H H ⊗n n! a! n!

and

√

2

a!Φa = a!. 2

As a consequence, the space L (Ω, G,√P ) is isometric to the Fock space, ∞ deﬁned as the orthogonal sum n=0 n!H ⊗n . In the next section we will 2 2 see that if H is an L space of the form L (T, B, µ), then In coincides with a multiple stochastic integral. 2

1.1.2 The white noise case: Multiple Wiener-Ito ˆ integrals Assume that the underlying separable Hilbert space H is an L2 space of the form L2 (T, B, µ), where (T, B) is a measurable space and µ is a σ-ﬁnite measure without atoms. In that case the Gaussian process W is characterized by the family of random variables {W (A), A ∈ B, µ(A) < ∞}, where W (A) = W (1A ). We can consider W (A) as an L2 (Ω, F, P )-valued measure on the parameter space (T, B), which takes independent values on any family of disjoint subsets of T , and such that any random variable W (A) has the distribution N (0, µ(A)) if µ(A) < ∞. We will say that W is an L2 (Ω)-valued Gaussian measure (or a Brownian measure) on (T, B). This measure will be also called the white noise based on µ. In that sense, W (h) can be regarded as the stochastic integral (Wiener integral) of the function h ∈ L2 (T ) with respect to W . We will write W (h) = T hdW , and observe that this stochastic integral cannot be deﬁned pathwise, because the paths of {W (A)} are not σ-additive measures on T . More generally, we will see in this section that the elements of the nth Wiener chaos Hn can be expressed as multiple stochastic integrals with respect to W . We start with the construction of multiple stochastic integrals. Fix m ≥ 1. Set B0 = {A ∈ B : µ(A) < ∞}. We want to deﬁne the multiple stochastic integral Im (f ) of a function f ∈ L2 (T m , B m , µm ). We denote by Em the set of elementary functions of the form f (t1 , . . . , tm ) =

n

ai1 ···im 1Ai1 ×···×Aim (t1 , . . . , tm ),

(1.10)

i1 ,...,im =1

where A1 , A2 , . . . , An are pairwise-disjoint sets belonging to B0 , and the coeﬃcients ai1 ···im are zero if any two of the indices i1 , . . . , im are equal.

1.1 Wiener chaos and stochastic integrals

9

The fact that f vanishes on the rectangles that intersect any diagonal subspace {ti = tj , i = j} plays a basic role in the construction of the multiple stochastic integral. For a function of the form (1.10) we deﬁne n

Im (f ) =

ai1 ···im W (Ai1 ) · · · W (Aim ).

i1 ,...,im =1

This deﬁnition does not depend on the particular representation of f , and the following properties hold: (i) Im is linear, (ii) Im (f ) = Im (f), where f denotes the symmetrization of f , which means 1 f(t1 , . . . , tm ) = f (tσ(1) , . . . , tσ(m) ), m! σ σ running over all permutations of {1, . . . , m}, 0 if m = q, (iii) E(IIm (f )IIq (g)) = m!f, gL2 (T m ) if m = q. Proof of these properties: Property (i) is clear. In order to show (ii), by linearity we may assume that f (t1 , . . . , tm ) = 1Ai1 ×···×Aim (t1 , . . . , tm ), and in this case the property is immediate. In order to show property (iii), consider two symmetric functions f ∈ Em and g ∈ Eq . We can always assume that they are associated with the same partition A1 , . . . , An . The case m = q is easy. Finally, let m = q and suppose that the functions f and g are given by (1.10) and by n

g(t1 , . . . , tm ) =

bi1 ···im 1Ai1 ×···×Aim (t1 , . . . , tm ),

i1 ,...,im =1

respectively. Then we have E(IIm (f )IIm (g))

= E

×

m! ai1 ···im W (Ai1 ) · · · W (Aim )

i1 <···

m! bi1 ···im W (Ai1 ) · · · W (Aim )

i1 <···
=

(m!)2 ai1 ···im bi1 ···im µ(Ai1 ) · · · µ(Aim )

i1 <···
= m!f, gL2 (T m ) . In order to extend the multiple stochastic integral to the space L2 (T m ), we have to prove that the space Em of elementary functions is dense in

10

1. Analysis on the Wiener space

L2 (T m ). To do this it suﬃces to show that the characteristic function of any set A = A1 × A2 × · · · × Am , Ai ∈ B0 , 1 ≤ i ≤ m, can be approximated by elementary functions in Em . Using the nonexistence of atoms for the measure µ, for any > 0 we can determine a system of pairwise-disjoint sets {B1 , . . . , Bn } ⊂ B0 , such that µ(Bi ) < for any i = 1, . . . , n, and each Ai can be expressed as the disjoint union of some of the Bj . This is possible because for any set A ∈ B0 of measure diﬀerent from zero and any 0 < γ < µ(A) we can ﬁnd a measurable set B ⊂ A of measure γ. Set µ(∪m i=1 Ai ) = α. We have 1A =

n

i1 ···im 1Bi1 ×···×Bim ,

i1 ,...,im =1

where i1 ···im is 0 or 1. We divide this sum into two parts. Let I be the set of mples (i1 , . . . , im ), where all the indices are diﬀerent, and let J be the set of the remaining mples. We set 1B = i1 ···im 1Bi1 ×···×Bim . (i1 ,...,im )∈I

Then 1B belongs to the space Em , B ⊂ A, and we have 1A − 1B 2L2 (T m ) = i1 ···im µ(Bi1 ) · · · µ(Bim ) (i1 ,...,im )∈J

≤ ≤

m 2 m 2

n

2

µ(Bi )

i=1

n

m−2 µ(Bi )

i=1

αm−1 ,

which shows the desired approximation. Letting f = g in property (iii) obtains E(IIm (f )2 ) = m!f2L2 (T m ) ≤ m! f 2L2 (T m ) . Therefore, the operator Im can be extended to a linear and continuous operator from L2 (T m ) to L2 (Ω, F, P ), which satisﬁes properties (i), (ii), and (iii). We will also write T m f (t1 , . . . , tm )W (dt1 ) · · · W (dtm ) for Im (f ). If f ∈ L2 (T p ) and g ∈ L2 (T q ) are symmetric functions, for any 1 ≤ r ≤ min(p, q) the contraction of r indices of f and g is denoted by f ⊗r g and is deﬁned by (f ⊗r g)(t1 , . . . , tp+q−2r ) = f (t1 , . . . , tp−r , s)g(tp+1 , . . . , tp+q−r , s)µr (ds). Tr

Notice that f ⊗r g ∈ L2 (T p+q−2r ).

1.1 Wiener chaos and stochastic integrals

11

The tensor product f ⊗ g and the contractions f ⊗r g, 1 ≤ r ≤ min(p, q), are not necessarily symmetric even though f and g are symmetric. We will and f ⊗ r g, respectively. denote their symmetrizations by f ⊗g The next formula for the multiplication of multiple integrals will play a basic role in the sequel. Proposition 1.1.2 Let f ∈ L2 (T p ) be a symmetric function and let g ∈ L2 (T ). Then, Ip (f )I1 (g) = Ip+1 (f ⊗ g) + pIIp−1 (f ⊗1 g).

(1.11)

Proof: By the density of elementary functions if L2 (T p ) and by linearity we can assume that f is the symmetrization of the characteristic function of A1 × · · · × Ap , where the Ai are pairwise-disjoint sets of B0 , and g = 1A1 or 1A0 , where A0 is disjoint with A1 , . . . , Ap . The case g = 1A0 is immediate because the tensor product f ⊗ g belongs to Ep+1 , and f ⊗1 g = 0. So, we assume g = 1A1 . Set β = µ(A1 ) · · · µ(Ap ). Given > 0, we can consider a measurable partition A1 = B1 ∪ · · · ∪ Bn such that µ(Bi ) < . Now we deﬁne the elementary function h = 1Bi ×Bj ×A2 ×···×Ap . i= j

Then we have Ip (f )I1 (g)

= W (A1 )2 W (A2 ) · · · W (Ap ) W (Bi )W (Bj )W (A2 ) · · · W (Ap ) = i= j

+

n

(W (Bi )2 − µ(Bi ))W (A2 ) · · · W (Ap )

i=1

+µ(A1 )W (A2 ) · · · W (Ap ) = Ip+1 (h ) + R + pIIp−1 (f ⊗1 g). Indeed, f ⊗1 g =

1 1A ×···×Ap µ(A1 ). p 2

We have 2L2 (T p+1 ) h − f ⊗g

A1 ×A1 ×A2 ×···×Ap 2L2 (T p+1 ) = h − 1 ≤ =

h − 1A1 ×A1 ×A2 ×···×Ap 2L2 (T p+1 ) n i=1

µ(Bi )2 µ(A2 ) · · · µ(Ap ) ≤ β

(1.12)

12

1. Analysis on the Wiener space

and E(R2 ) = 2

n

µ(Bi )2 µ(A2 ) · · · µ(Ap ) ≤ 2β,

i=1

and letting tend to zero in (1.12) we obtain the desired result. Formula (1.11) can be generalized as follows.

Proposition 1.1.3 Let f ∈ L2 (T p ) and g ∈ L2 (T q ) be two symmetric functions. Then p∧q p q Ip (f )IIq (g) = r! (1.13) Ip+q−2r (f ⊗r g). r r r=0 Proof: The proof can be done by induction with respect to the index q. We will assume that p ≥ q. For q = 1 it reduces to (1.11). Suppose it holds for q − 1. By a density argument we can assume that the function g is of 2 , where g1 and g2 are symmetric functions of q − 1 and the form g = g1 ⊗g one variable, respectively, such that g1 ⊗1 g2 = 0. By (1.11) we have 2 ) = Iq−1 (g1 )I1 (g2 ). Iq (g1 ⊗g Thus by the induction hypothesis, and using (1.11), we obtain Ip (f )IIq (g)

= Ip (f )IIq−1 (g1 )I1 (g2 ) q−1 p q−1 r! = Ip+q−1−2r (f ⊗r g1 )I1 (g2 ) r r r=0 q−1 p q−1 r g1 ) ⊗ g2 ) Ip+q−2r ((f ⊗ = r! r r r=0

r g1 ) ⊗1 g2 ) + (p + q − 1 − 2r)IIp+q−2r−2 ((f ⊗ q−1 p q−1 r g1 ) ⊗ g2 ) r! Ip+q−2r ((f ⊗ = r r r=0 q p q−1 (r − 1)! + r−1 r−1 r=1 r−1 g1 ) ⊗1 g2 ). × (p + q − 2r + 1)IIp+q−2r ((f ⊗

For any 1 ≤ r ≤ q, one can show the following equality: r g) = q(f ⊗

r(p + q − 2r + 1) r−1 g1 )⊗1 g2 +(q−r)((f ⊗ r g1 )⊗g 2 ). (1.14) (f ⊗ p−r+1

Substituting (1.14) into the above summations yields (1.13). The next result gives the relationship between Hermite polynomials and multiple stochastic integrals.

1.1 Wiener chaos and stochastic integrals

13

Proposition 1.1.4 Let Hm (x) be the mth Hermite polynomial, and let h ∈ H = L2 (T ) be an element of norm one. Then it holds that m! Hm (W (h)) = h(t1 ) · · · h(tm )W (dt1 ) · · · W (dtm ). (1.15) Tm

As a consequence, the multiple integral Im maps L2 (T m ) onto the Wiener chaos Hm . Proof: Eq. (1.15) will be proved by induction on m. For m = 1 it is immediate. Assume it holds for 1, 2, . . . , m. Using the recursive relation for the Hermite polynomials (1.3) and the product formula (1.11), we have Im+1 (h⊗(m+1) ) = Im (h⊗m )I1 (h) − mIIm−1 h⊗(m−1) h(t)2 µ(dt) T

= m! Hm (W (h))W (h) − m(m − 1)! Hm−1 (W (h)) = m!(m + 1)H Hm+1 (W (h)) = (m + 1)! Hm+1 (W (h)), where h⊗m denotes the function of m variables deﬁned by h⊗m (t1 , . . . , tm ) = h(t1 ) · · · h(tm ). Denote by L2S (T m ) the closed subspace of L2 (T m ) formed by symmetric functions. The multiple integral Im veriﬁes E(IIm (f )2 ) = m! f 2L2 (T m ) on L2S (T m ). So the image Im (L2S (T m )) is closed, and by (1.15) it contains the random variables Hm (W (h)), h ∈ H, and hH = 1. Consequently, Hm ⊂ Im (L2S (T m )). Due to the orthogonality between multiple integrals of diﬀerent order, we have that Im (L2S (T m )) is orthogonal to Hn , n = m. So, Im (L2S (T m )) = Hm , which completes the proof of the proposition. As a consequence we deduce the following version of the Wiener chaos expansion. Theorem 1.1.2 Any square integrable random variable F ∈ L2 (Ω, G, P ) (recall that G denotes the σ-ﬁeld generated by W ) can be expanded into a series of multiple stochastic integrals: F =

∞

In (ffn ).

n=0

Here f0 = E(F ), and I0 is the identity mapping on the constants. Furthermore, we can assume that the functions fn ∈ L2 (T n ) are symmetric and, in this case, uniquely determined by F . Let {ei , i ≥ 1} be an orthonormal basis of H, and ﬁx a miltiindex a = (a1 , . . . , aM , 0, . . .) such that |a| = a1 + · · · + aM = n. From (1.15) and

14

1. Analysis on the Wiener space

(1.13) it follows that a!

M i=1

Hai (W (ei ))

=

M

i Iai (e⊗a ) i

i=1

M 1 2 . ⊗ e⊗a ⊗ · · · ⊗ e⊗a = In e⊗a 1 2 M

Hence, the multiple stochastic integral In coincides with the isometry be (equipped with the modiﬁed tween √the symmetric tensor product H ⊗n norm n! ·H ⊗n ) and the nth Wiener chaos Hn introduced in (1.9). No tice that H ⊗n is isometric to L2S (T n ). Example 1.1.2 Suppose that the parameter space is T = R+ × {1, . . . , d} and that the measure µ is the product of the Lebesgue measure times the uniform measure, which gives mass one to each point 1, 2, . . . , d. Then we have H = L2 (R+ × {1, . . . , d}, µ) ∼ = L2 (R+ ; Rd ). In this situation we have i that W (t) = W ([0, t] × {i}), 0 ≤ t ≤ 1, 1 ≤ i ≤ d, is a standard ddimensional Brownian motion. That is, {W i (t), t ∈ R+ }, i = 1, . . . , d, are independent zero-mean Gaussian processes with covariance function E(W i (s)W i (t)) = s ∧ t. Furthermore, for any h ∈ H, the random vari d ∞ Wti . able W (h) can be obtained as the stochastic integral i=1 0 hit dW The Brownian motion veriﬁes E(|W i (t) − W i (s)|2 ) = |t − s| for any s, t ≥ 0, i = 1, . . . , d. This implies that (2k)! |t − s|k 2k k! for any integer k ≥ 2. From Kolmogorov’s continuity criterion (see the appendix, Section A.3) it follows that W possesses a continuous version. Consequently, we can deﬁne the d-dimensional Brownian motion on the canonical space Ω = C0 (R+ ; Rd ). The law of the process W is called the Wiener measure. E(|W i (t) − W i (s)|2k ) =

In this example multiple stochastic integrals can be considered as iterated Ito ˆ stochastic integrals with respect to the Brownian motion, as we shall see in the next section. Example 1.1.3 Take T = R2+ and µ equal to the Lebesgue measure. Let W be a white noise on T . Then W (s, t) = W ([0, s] × [0, t]), s, t ∈ R+ , deﬁnes a two-parameter, zero-mean Gaussian process with covariance given by E(W (s, t)W (s , t )) = (s ∧ s )(t ∧ t ), which is called the Wiener sheet or the two-parameter Wiener process. The process W has a version with continuous paths. This follows easily from Kolmogorov’s continuity theorem, taking into account that E(|W (s, t) − W (s , t )|2 ) ≤ max(s, s , t, t )(|s − s | + |t − t |).

1.1 Wiener chaos and stochastic integrals

15

1.1.3 Itˆˆo stochastic calculus In this section we survey some of the basic properties of the stochastic integral of adapted processes with respect to the Brownian motion, introduced by Itˆ o. Suppose that W = {W (t), t ≥ 0} is a standard Brownian motion deﬁned on the canonical probability space (Ω, F, P ). That is, Ω = C0 (R+ ) and P is a probability measure on the Borel σ-ﬁeld B(Ω) such that the canonical process Wt (ω) = ω(t) is a zero-mean Gaussian process with covariance E(W Ws Wt ) = s ∧ t. The σ-ﬁeld F will be the completion of B(Ω) with respect to P . We know that the sequence Sn (t) = [W (tk2−n ) − W (t(k − 1)2−n )]2 1≤k≤2n

converges almost surely and in L2 (Ω) to the constant t, as n tends to inﬁnity. In other words, the paths of the Brownian motion have a quadratic variation equal to t. This property, together with the continuity of the paths, implies that the paths of W have inﬁnite total variation on any bounded interval. Consequently, we cannot deﬁne path-wise a stochastic integral of the form t u(s)W (ds), 0

where u = {u(t), t ≥ 0} is a given stochastic process. If the paths of the process u have ﬁnite total variation on bounded intervals, we can overcome this diﬃculty by letting

t

u(s)W (ds) = u(t)W (t) − 0

t

W (s)u(ds). 0

However, most of the processes that we will ﬁnd (like W itself) do not have paths with ﬁnite total variation on bounded intervals. For each t ≥ 0 we will denote by Ft the σ-ﬁeld generated by the random variables {W (s), 0 ≤ s ≤ t} and the null sets of F. Then a stochastic process u = {u(t), t ≥ 0} will be called adapted or nonanticipative if u(t) is Ft -measurable for any t ≥ 0. We will ﬁx a time interval, denoted by T , which can be [0, t0 ] or R+ . We will denote by L2 (T × Ω) = L2 (T × Ω, B(T ) ⊗ F, λ1 × P ) (where λ1 denotes the Lebesgue measure) the set of square integrable processes, and L2a (T × Ω) will represent the subspace of adapted processes. Let E be the class of elementary adapted processes. That is, a process u belongs to E if it can be written as u(t) =

n i=1

Fi 1(ti ,ti+1 ] (t),

(1.16)

16

1. Analysis on the Wiener space

where 0 ≤ t1 < · · · < tn+1 are points of T , and every Fi is an Fti measurable and square integrable random variable. Then we have the following result. Lemma 1.1.3 The class E is dense in L2a (T × Ω). Proof: Suppose T = [0, 1]. Let u be a process in L2a (T × Ω), and consider the sequence of processes deﬁned by

u (t) = n

n 2 −1

i2−n

n

2

(i−1)2−n

i=1

u(s)ds 1(i2−n ,(i+1)2−n ] (t).

(1.17)

We claim that the sequence u n converges to u in L2 (T × Ω). In fact, deﬁne n . Then Pn is a linear operator in L2 (T × Ω) with norm bounded Pn (u) = u by one, such that Pn (u) → u as n tends to inﬁnity whenever the process u is continuous in L2 (Ω). The proof now follows easily. Remark: A measurable process u : T × Ω → R is called progressively measurable if the restriction of u to the product [0, t] × Ω is B([0, t]) ⊗ Ft measurable for all t ∈ T . One can show (see [225, Theorem 4.6]) that any adapted process has a progressively measurable version, and we will always assume that we are dealing with this kind of version. This is necessary, for instance, to ensure that the approximating processes u n introduced in Lemma 1.1.3 are adapted. For a nonanticipating process of the form (1.16), the random variable u(t)dW Wt = T

n

Fi (W (ti+1 ) − W (ti ))

(1.18)

i=1

will be called the stochastic integral (or the Itˆ o integral) of u with respect to the Brownian motion W . The Ito ˆ integral of elementary processes is a linear functional that takes values on L2 (Ω) and has the following basic properties: E( u(t)dW Wt ) = 0,

(1.19)

T

u(t)dW Wt |2 ) = E( u(t)2 dt).

E(| T

(1.20)

T

Property (1.19) is immediate from both (1.18) and the fact that for each i = 1, . . . , n the random variables Fi and W (ti+1 ) − W (ti ) are independent.

1.1 Wiener chaos and stochastic integrals

17

Proof of (1.20): We have u(t)dW Wt |2 )

E(| T

=

n

E(|F Fi (W (ti+1 ) − W (ti ))|2 )

i=1

+2

E(F Fi Fj (W (ti+1 ) − W (ti ))

i<j

× (W (tj+1 ) − W (tj ))) n 2 = E(F Fi )(ti+1 − ti ) = E( u(t)2 dt), T

i=1

because whenever i < j, W (tj+1 )−W (tj ) is independent of Fi Fj (W (ti+1 )− W (ti )). The isometry property (1.20) allows us to extend the Itˆ oˆ integral to the class L2a (T × Ω) of adapted square integrable processes, and the above properties still hold in this class. The Ito ˆ integral veriﬁes the following local property: u(t)dW Wt = 0, T

almost surely (a.s.) on the set G = { T u(t)2 dt = 0}. In fact, onnthe set G (t)dW Wt = the processes { un } introduced in (1.17) vanish, and therefore T u n (t)dW Wt to 0 on G. Then the result follows from the convergence of T u u(t)dW Wt in L2 (Ω). T We also have for any u ∈ L2a (T × Ω), > 0, and K > 0, K P u(t)dW Wt > ≤ P u(t)2 dt > K + 2 . T

(1.21)

T

Proof of (1.21): Deﬁne u (t) = u(t)1{ t u(s)2 ds≤K} . 0

× Ω), and using the local property of the The process u belongs to Ito ˆ integral we obtain 2 P u(t)dW Wt > , u(t) dt ≤ K T T 2 =P u (t)dW Wt > , u(t) dt ≤ K T T 1 K 2 ≤P u (t)dW Wt > ≤ 2 E u (t) dt ≤ 2 . T T L2a (T

18

1. Analysis on the Wiener space

Using property (1.21), one can extend the Itˆoˆ integral to the class of measurable and adapted processes such that u(t)2 dt < ∞ a.s., T

and the local property still holds for these processes. Suppose that u belongs to L2a (T × Ω). Then the indeﬁnite integral

t

u(s)dW Ws = 0

T

u(s)1[0,t] (s)dW Ws ,

t ∈ T,

is a martingale with respect to the increasing family of σ-ﬁelds {F Ft , t ≥ 0}. Indeed, the martingale property is easy to check for elementary processes and is transferred to general adapted processes by L2 convergence. If u is an elementary process of the form (1.16), the martingale

t

u(s)dW Ws = 0

n

Fi (W (ti+1 ∧ t) − W (ti ∧ t))

i=1

clearly of a continuous version yt possesses a continuous version. The existence Ws } in the general case u ∈ L2a (T × Ω) follows from Doob’s for { 0 u(s)dW maximal inequality for martingales (see (A.2)) and from the Borel-Cantelli lemma. If u is an adapted and measurable process such that T u(t)2 dt < ∞, then the indeﬁnite integral is a continuous local martingale. That is, if we deﬁne the random times t Tn = inf{t ≥ 0 : u(s)2 ds ≥ n}, n ≥ 1, 0

then: (i) For each n ≥ 1, Tn is a stopping time (i.e., {T Tn ≤ t} ∈ Ft for any t ≥ 0). (ii) Tn ↑ ∞ as n tends to inﬁnity. (iii) The processes

Mn (t) = 0

t

u(s)1{s≤TTn } dW Ws

are continuous square integrable martingales such that t u(s)dW Ws Mn (t) = 0

whenever t ≤ Tn . In fact, u1[0,TTn ] ∈ L2a (T × Ω) for each n.

1.1 Wiener chaos and stochastic integrals

19

Let u be an adapted and measurable process such that T u(t)2 dt < ∞, t and consider the continuous local martingale M (t) = 0 u(s)dW Ws . Deﬁne M t =

t

u(s)2 ds. 0

Then Mt2 − M t is a martingale when u ∈ L2a (T × Ω). This is clear if u is an elementary process of the form (1.16), and in the general case it holds by approximation. The increasing process M t is called the quadratic variation of the local

n−1 Mti+1 − Mti )2 , when π = {0 = martingale M . That is, the family i=0 (M t0 < t1 < · · · < tn = t} runs over all the partitions of [0, t], converges t in probability to 0 u(s)2 ds as |π| = maxi (ti+1 − ti ) tends to zero. Indeed, by a localization argument, it suﬃces to prove the convergence when t 2 u(s) ds ≤ K for some constant K > 0, and in this case it holds in L2 (Ω) 0 due to Burkholder’s inequality (A.3) and the fact that Mt2 − M t is a square integrable martingale. In fact, we have ⎛ 2 ⎞ ti+1 n−1 u2 (s)ds − (M Mti+1 − Mti )2 ⎠ E ⎝ t i i=0 2 ti+1 n−1 2 2 = E u (s)ds − (M Mti+1 − Mti ) ti i=0 2 s n−1 ti+1 2 2 ≤c E u (s)ds ≤ cKE sup u (θ)dθ i=0

|s−r|≤|π|

ti

r

for some constant c > 0, and this converges to zero as |π| tends to zero. One of the most important tools in the stochastic calculus is the changeof-variable formula, or Ito’s ˆ formula. Proposition 1.1.5 Let F : R → R be a twice continuously diﬀerentiable function. τ measurable and adapted processes ver τ Suppose that u and v are ifying 0 u(t)2 dt < ∞ a.s. and 0 |v(t)|dt < ∞ a.s. for every τ ∈ T . Set t t X(t) = X(0) + 0 u(s)dW Ws + 0 v(s)ds. Then we have F (Xt ) − F (X0 ) =

t F (Xs )us dW Ws + F (Xs )vs ds 0 0 1 t F (Xs )u2s ds. + 2 0 t

(1.22)

The proof of (1.22) comes from the fact that the quadratic variation t of the process X(t) is equal to 0 u2s ds; consequently, when we develop by Taylor’s expansion the function F (X(t)), there is a contribution from

20

1. Analysis on the Wiener space

the second-order term, which produces the additional summand in Itˆ oˆ’s formula. Proof:

By a localization procedure we can assume F ∈ Cb2 (R) and 2 |v(t)|dt} ≤ K sup{ u(t) dt, T

T

for some constant K > 0. Fix t > 0. For any partition π = {0 = t0 < t1 < · · · < tn = t} we can write, using Taylor’s formula, n−1

F (Xt ) − F (X0 ) =

(F (Xti+1 ) − F (Xti ))

i=0 n−1

=

F (Xti )(Xti+1 − Xti )

i=0

+

n−1 1 F (X i )(Xti+1 − Xti )2 , 2 i=0

where X i is a random point between Xti and Xti+1 . The ﬁrst summand t t in the above expression converges to 0 F (Xs )us dW Ws + 0 F (Xs )vs ds in L2 (Ω) as |π| = maxi (ti+1 − ti ) tends to zero. For the second summand we use the decomposition (Xti+1 − Xti )2

us dW Ws vs ds

ti+1

us dW Ws

vs ds

ti

2

ti+1

+

ti+1

+2

ti

2

ti+1

=

ti

.

ti

Only the ﬁrst term produces a nonzero contribution. Then we can write

t

F

(Xs )u2s ds

−

0

=

n−1 ti+1 i=0

+

ti

n−1

+

2

ti+1

F (X i )

us dW Ws ti

i=0

[F (Xs ) − F (Xti )] u2s ds

F (Xti )

i=0 n−1

n−1

ti+1

u2s ds

ti

[F (Xti ) − F (X i )]

i=0

= a1 + a2 + a3 .

−

2

ti+1

us dW Ws ti ti+1

us dW Ws ti

2

1.1 Wiener chaos and stochastic integrals

21

We have |a1 | ≤ K

|a3 | ≤

sup |s−r|≤|π|

sup |s−r|≤|π|

|F (Xs ) − F (Xr )| , n−1

|F (Xs ) − F (Xr )|

2

ti+1

us dW Ws

.

ti

i=0

These expressions converge to zero in probability as |π| tends to zero. Finally, applying Burkholder’s inequality (A.3) and the martingale property t t Ws )2 − 0 u2s ds, we can get a constant c > 0 such that of ( 0 us dW ⎛ E(|a2 |2 )

= E⎝

n−1

F (Xti )2

i=0

≤ cF ∞

n−1 i=0

≤ KcF ∞ E

ti+1

u2s ds −

ti

ti

us dW Ws

⎞ ⎠

u2s ds

s

u2θ dθ

sup |s−r|≤|π|

2 2

ti+1

ti

2

ti+1

E

,

r

and this converges to zero as |π| tends to zero.

Consider two adapted processes {ut , t ∈ T } and {vt , t ∈ T } such that t u(s)2 ds < ∞ a.s. and 0 |v(s)|ds < ∞ a.s. for all t ∈ T . Let X0 ∈ R. 0 The process t t Xt = X0 + us dW Ws + vs ds (1.23)

t

0

0

t t Ws and Vt = 0 vs ds is called a continuous semimartingale, and Mt = 0 us dW are the local martingale part and bounded variation part of X, respectively. Ito’s ˆ formula tells us that this class of processes is stable by the composition with twice continuously diﬀerentiable functions. Let π = {0 = t0 < t1 < · · · < tn = t} be a partition of the interval [0, t]. The sums n−1 1 (Xti + Xti+1 )(W Wti+1 − Wti ) (1.24) 2 i=0 converge in probability as |π| tends to zero to

t

Xs dW Ws + 0

1 2

t

us ds. 0

This expression is called the Stratonovich integral of X with respect to W t and is denoted by 0 Xs ◦ dW Ws .

22

1. Analysis on the Wiener space

The convergence of the sums in (1.24) follows easily from the decomposition 1 (Xti + Xti+1 )(W Wti+1 − Wti ) 2

= Xti (W Wti+1 − Wti ) 1 Wti+1 − Wti ), + (Xti+1 − Xti )(W 2

and the fact that the joint quadratic t variation of the processes X and W (denoted by X, W t ) is equal to 12 0 us ds. t t Ws − 12 0 u2s ds). As an Let u ∈ L2a (T × Ω). Set Mu (t) = exp( 0 us dW application of Itˆ oˆ’s formula we deduce t Mu (t) = 1 + Mu (s)u(s)dW Ws . (1.25) 0

That means Mu is a local martingale. In particular, if u = h is a deterministic square integrable function of the space H = L2 (T ), then Mh is a square integrable martingale. Formula (1.25) shows that exp(W Wt − 2t ) plays the role of the customary exponentials in the stochastic calculus. The following result provides an integral representation of any square functional of the Brownian motion. Set FT = σ{W (s), s ∈ T }. Theorem 1.1.3 Let F be a square integrable random variable. Then there exists a unique process u ∈ L2a (T × Ω) such that ut dW Wt . (1.26) F = E(F ) + T

Proof: To prove the theorem it suﬃces to show that any zero-mean square integrable random variable G that is orthogonal to all the stochastic Wt , u ∈ L2a (T × Ω) must be zero. In view of formula (1.25), integrals T ut dW such a random variable G is orthogonal to the exponentials 1 E(h) = exp( hs dW Ws − h2s ds), 2 T T h ∈ L2 (T ). Finally, because these exponentials form a total subset of L2 (Ω, FT , P ) by Lemma 1.1.2, we can conclude this proof. As a consequence of this theorem, any square integrable martingale on the time interval T can be represented as an indeﬁnite Ito ˆ integral. In fact, Ft ) for some random variable such a martingale has the form Mt = E(F |F F ∈ L2 (Ω, FT , P ). Then, taking conditional expectations with respect to the σ-ﬁeld Ft in Eq. (1.26), we obtain E(F |F Ft ) = E(F ) +

t

us dW Ws . 0

1.1 Wiener chaos and stochastic integrals

23

Let fn : T n → R be a symmetric and square integrable function. For these functions the multiple stochastic integral In (ffn ) with respect to the Ws , h ∈ L2 (T )} introduced in Section Gaussian process {W (h) = T hs dW 1.1.2 coincides with an iterated Itoˆ integral. That is, assuming T = R+ , we have ∞ tn t2 In (ffn ) = n! ··· fn (t1 , . . . , tn )dW Wt1 · · · dW Wtn . (1.27) 0

0

0

Indeed, this equality is clear if fn is an elementary function of the form (1.10), and in the general case the equality will follow by a density argument, taking into account that the iterated stochastic Itˆ oˆ integral veriﬁes the same isometry property as the multiple stochastic integral. Let {W (t), t ≥ 0} be a d-dimensional Brownian motion. In this case the multiple stochastic integral In (ffn ) is deﬁned for square integrable kernels fn ((t1 , i1 ), . . . , (tn , in )), which are symmetric in the variables (tj , ij ) ∈ R+ × {1, . . . , d}, and it can be expressed as a sum of iterated Itˆo integrals: In (ffn )

= n!

d

i1 ,...,in =1

∞

0

tn

···

0

t2

fn ((t1 , i1 ), . . . , (tn , in )) 0

× dW Wti11 · · · dW Wtinn .

Exercises 1.1.1 For every n let us deﬁne the Hermite polynomial Hn (λ, x) by n x Hn (λ, x) = λ 2 Hn ( √ ), where x ∈ R and λ > 0. λ

Check that exp(tx −

∞ t2 λ )= tn Hn (λ, x). 2 n=0

Let W be a white noise on a measure space (T, B, µ). Show that Hm (h2H , W (h)) =

1 Im (h⊗m ) m!

for any h ∈ L2 (T, B, µ). 1.1.2 Using the recursive formula (1.2), deduce the following explicit expression for the Hermite polynomials

[n/2]

Hn (x) =

k=0

(−1)k xn−2k . k ! (n − 2k)! 2k

24

1. Analysis on the Wiener space

As an application show that if Y is a random variable with distribution N (0, σ 2 ), then (σ 2 − 1)m , E(H H2m (Y )) = 2m m! and E(H Hn (Y )) = 0 if n is odd. 1.1.3 Let {W Wt , t ≥ 0} be a one-dimensional Brownian motion. Show that the process {H Hn (t, Wt ), t ≥ 0} (where Hn (t, x) is the Hermite polynomial introduced in Exercise 1.1.1) is a martingale. 1.1.4 Let W = {W (h), h ∈ H} be an isonormal Gaussian process deﬁned on the probability space (Ω, F, P ), where F is generated by W . Let V be a real separable Hilbert space. Show the Wiener chaos expansion L2 (Ω; V ) =

∞

Hn (V ),

n=0 2 where Hn (V ) is the closed subspace

m of L (Ω; V ) generated by the V -valued random variables of the form j=1 Fj vj , Fj ∈ Hn and vj ∈ V . Construct

an isometry between H ⊗n ⊗ V and Hn (V ) as in (1.9).

1.1.5 By iteration of the representation formula (1.26) and using expression (1.27) show that any random variable F ∈ L2 (Ω, F, P ) (where F is generated by W ) can be expressed as an inﬁnite sum of orthogonal multiple stochastic integrals. This provides an alternative proof of the Wiener chaos expansion for Brownian functionals. 1.1.6 Prove Eq. (1.14). 1.1.7 Let us denote by P the family of random variables of the form p(W (h1 ), . . . , W (hn )), where hi ∈ H and p is a polynomial. Show that P is dense in Lr (Ω) for all r ≥ 1. Hint: Assume that r > 1 and let q be the conjugate of r. As in the proof of Theorem 1.1.1 show that if Z ∈ Lq (Ω) veriﬁes E(ZY ) = 0 for all Y ∈ P, then Z = 0.

1.2 The derivative operator This section will be devoted to the properties of the derivative operator. Let W = {W (h), h ∈ H} denote an isonormal Gaussian process associated with the Hilbert space H. We assume that W is deﬁned on a complete probability space (Ω, F, P ), and that F is generated by W . We want to introduce the derivative DF of a square integrable random variable F : Ω → R. This means that we want to diﬀerentiate F with respect to the chance parameter ω ∈ Ω. In the usual applications of this theory, the space Ω will be a topological space. For instance, in the example

1.2 The derivative operator

25

of the d-dimensional Brownian motion, Ω is the Fr´ ´echet space C0 (R+ ; Rd ). However, we will be interested in random variables F that are deﬁned P a.s. and that do not possess a continuous version (see Exercise 1.2.1). For this reason we will introduce a notion of derivative deﬁned in a weak sense, and without assuming any topological structure on the space Ω. We denote by Cp∞ (Rn ) the set of all inﬁnitely continuously diﬀerentiable functions f : Rn → R such that f and all of its partial derivatives have polynomial growth. Let S denote the class of smooth random variables such that a random variable F ∈ S has the form F = f (W (h1 ), . . . , W (hn )),

(1.28)

where f belongs to Cp∞ (Rn ), h1 , . . . , hn are in H, and n ≥ 1. ∂f We will make use of the notation ∂i f = ∂x and ∇f = (∂1 f, . . . , ∂n f ), i 1 n whenever f ∈ C (R ). We will denote by Sb and S0 the classes of smooth random variables of the form (1.28) such that the function f belongs to Cb∞ (Rn ) (f and all of its partial derivatives are bounded) and to C0∞ (Rn ) (f has compact support), respectively. Moreover, we will denote by P the class of random variables of the form (1.28) such that f is a polynomial. Note that P ⊂ S, S0 ⊂ Sb ⊂ S, and that P and S0 are dense in L2 (Ω). Deﬁnition 1.2.1 The derivative of a smooth random variable F of the form (1.28) is the H-valued random variable given by DF =

n

∂i f (W (h1 ), . . . , W (hn ))hi .

(1.29)

i=1

For example, DW (h) = h. In order to interpret DF as a directional derivative, note that for any element h ∈ H we have DF, hH

=

1 lim [f (W (h1 ) + h1 , hH , . . . , W (hn ) + hn , hH ) − f (W (h1 ), . . . , W (hn ))].

→0

Roughly speaking, the scalar product DF, hH is the derivative at = 0 of the random variable F composed with shifted process {W (g)+g, hH , g ∈ H}. The following result is an integration-by-parts formula that will play a fundamental role along this chapter. Lemma 1.2.1 Suppose that F is a smooth random variable and h ∈ H. Then E(DF, hH ) = E(F W (h)). (1.30)

26

1. Analysis on the Wiener space

Proof: First notice that we can normalize Eq. (1.30) and assume that the norm of h is one. There exist orthonormal elements of H, e1 , . . . , en , such that h = e1 and F is a smooth random variable of the form F = f (W (e1 ), . . . , W (en )), where f is in Cp∞ (Rn ). Let φ(x) denote the density of the standard normal distribution on Rn , that is, 1 2 x ). 2 i=1 i n

φ(x) = (2π)− 2 exp(− n

Then we have

E(DF, hH ) =

Rn

=

Rn

∂1 f (x)φ(x)dx f (x)φ(x)x1 dx

= E(F W (e1 )) = E(F W (h)), which completes the proof of the lemma. Applying the previous result to a product F G, we obtain the following consequence. Lemma 1.2.2 Suppose that F and G are smooth random variables, and let h ∈ H. Then we have E(GDF, hH ) = E(−F DG, hH + F GW (h)).

(1.31)

As a consequence of the above lemma we obtain the following result. Proposition 1.2.1 The operator D is closable from Lp (Ω) to Lp (Ω; H) for any p ≥ 1. Proof: Let {F FN , N ≥ 1} be a sequence of smooth random variables such FN that FN converges to zero in Lp (Ω) and the sequence of derivatives DF converges to η in Lp (Ω; H). Then, from Lemma 1.2.2 it follows that η is equal to zero. Indeed, for any h ∈ H and for any smooth random variable 2 F ∈ Sb such that F W (h) is bounded (for intance, F = Ge−εW (h) where G ∈ Sb and ε > 0), we have E(η, hH F )

= =

lim E(DF FN , hH F )

N →∞

lim E(−F FN DF, hH + FN F W (h)) = 0,

N →∞

because FN converges to zero in Lp (Ω) as N tends to inﬁnity, and the random variables DF, hH and F W (h) are bounded. This implies η = 0.

1.2 The derivative operator

27

For any p ≥ 1 we will denote the domain of D in Lp (Ω) by D1,p , meaning that D1,p is the closure of the class of smooth random variables S with respect to the norm 1

F 1,p = [E(|F |p ) + E(DF pH )] p . For p = 2, the space D1,2 is a Hilbert space with the scalar product F, G1,2 = E(F G) + E(DF, DGH ). We can deﬁne the iteration of the operator D in such a way that for a smooth random variable F , the iterated derivative Dk F is a random variable with values in H ⊗k . Then for every p ≥ 1 and any natural number k ≥ 1 we introduce the seminorm on S deﬁned by ⎡ ⎤ p1 k E(Dj F pH ⊗j )⎦ . (1.32) F k,p = ⎣E(|F |p ) + j=1

This family of seminorms veriﬁes the following properties: (i) Monotonicity:

F k,p ≤ F j,q , for any F ∈ S, if p ≤ q and k ≤ j.

(ii) Closability: The operator Dk is closable from S into Lp (Ω; H ⊗k ), for all p ≥ 1. Proof: 1.2.3).

The proof is analogous to the case where k = 1 (see Exercise

(iii) Compatibility: Let p, q ≥ 1 be real numbers and k, j be natural numbers. Suppose that Fn is a sequence of smooth random variables Fn − such that F Fn k,p converges to zero as n tends to inﬁnity, and F Fn j,q tends Fm j,q converges to zero as n, m tend to inﬁnity. Then F to zero as n tends to inﬁnity. Proof: This is an immediate consequence of the closability of the operators Di , i ≥ 1, on S. We will denote by Dk,p the completion of the family of smooth random variables S with respect to the norm · k,p . From property (i) it follows that Dk+1,p ⊂ Dk,q if k ≥ 0 and p > q. For k = 0 we put · 0,p = · p and D0,p = Lp (Ω). Fix an element h ∈ H. We can deﬁne the operator Dh on the set S of smooth random variables by Dh F = DF, hH .

(1.33) p

p

By Lemma 1.2.2 this operator is closable from L (Ω) into L (Ω), for any p ≥ 1, and it has a domain that we will denote by Dh,p . The following result characterizes the domain of the derivative operator D1,2 in terms of the Wiener chaos expansion.

28

1. Analysis on the Wiener space

Proposition 1.2.2 Let F be

a square integrable random variable with the ∞ Wiener chaos expansion F = n=0 Jn F . Then F ∈ D1,2 if and only if ∞

2

E(DF H ) =

2

n J Jn F 2 < ∞.

(1.34)

n=1

Moreover, if (1.34) holds, then for all n ≥ 1 we have D(J Jn F ) = Jn−1 (DF ). Proof: The derivative of a random variable of the form Φa , deﬁned in (1.6), can be computed using (1.2): D(Φa ) =

∞ ∞ √ a! Hai (W (ei ))H Haj −1 (W (ej ))ej . j=1 i=1,i= j

Then, D(Φa ) ∈ Hn−1 (H) (see Execise 1.1.4) if |a| = n, and ∞ 2 ∞ E D(Φa )H = j=1

a! = |a|. a i=1,i= j i !(aj − 1)!

The proposition follows easily from Proposition 1.1.1. Jn F ) = Jn−k (Dk F ) for all k ≥ 2 and n ≥ k. By iteration we obtain Dk (J Hence, ∞

2

2 n(n − 1) · · · (n − k + 1) J Jn F 2 , E( Dk F H ⊗k ) = n=k

and F ∈ Dk,2 if and only if

∞ n=1

2

nk J Jn F 2 < ∞.

The following result is the chain rule, which can be easily proved by approximating the random variable F by smooth random variables and the function ϕ by ϕ ∗ ψ , where {ψ } is an approximation of the identity. Proposition 1.2.3 Let ϕ : Rm → R be a continuously diﬀerentiable function with bounded partial derivatives, and ﬁx p ≥ 1. Suppose that F = (F 1 , . . . , F m ) is a random vector whose components belong to the space D1,p . Then ϕ(F ) ∈ D1,p , and D(ϕ(F )) =

m

∂i ϕ(F )DF i .

i=1

Let us prove the following technical result. Lemma 1.2.3 Let {F Fn , n ≥ 1} be a sequence of random variables in D1,2 that converges to F in L2 (Ω) and such that Fn 2H < ∞. sup E DF n

Fn , n ≥ 1} Then F belongs to D , and the sequence of derivatives {DF converges to DF in the weak topology of L2 (Ω; H). 1,2

1.2 The derivative operator

29

Proof: There exists a subsequence {F Fn(k) , k ≥ 1} such that the sequence of derivatives DF Fn(k) converges in the weak topology of L2 (Ω; H) to some Fn(k) on element α ∈ L2 (Ω; H). By Proposition 1.2.2, the projections of DF any Wiener chaos converge in the weak topology of L2 (Ω), as k tends to inﬁnity, to those of α. Consequently, Proposition 1.2.2 implies F ∈ D1,2 and α = DF . Moreover, for any weakly convergent subsequence the limit must be equal to α by the preceding argument, and this implies the weak convergence of the whole sequence. The chain rule can be extended to the case of a Lipschitz function: Proposition 1.2.4 Let ϕ : Rm → R be a function such that |ϕ(x) − ϕ(y)| ≤ K|x − y| for any x, y ∈ Rm . Suppose that F = (F 1 , . . . , F m ) is a random vector whose components belong to the space D1,2 . Then ϕ(F ) ∈ D1,2 , and there exists a random vector G = (G1 , . . . , Gm ) bounded by K such that D(ϕ(F )) =

m

Gi DF i .

(1.35)

i=1

Proof: If the function ϕ is continuously diﬀerentiable, then the result reduces to that of Proposition 1.2.3 with Gi = ∂i ϕ(F ). Let αn (x) be a sequence of regularization kernels of the form αn (x) = nm α(nx), where α is a nonnegative function belonging to C0∞ (Rm ) whose support is the unit ball and such that Rm α(x)dx = 1. Set ϕn = ϕ ∗ αn . It is easy to check that limn ϕn (x) = ϕ(x) uniformly with respect to x, and the functions ϕn are C ∞ with |∇ϕn | ≤ K. For each n we have D(ϕn (F )) =

m

∂i ϕn (F )DF i .

(1.36)

i=1

The sequence ϕn (F ) converges to ϕ(F ) in L2 (Ω) as n tends to inﬁnity. On the other hand, the sequence {D(ϕn (F )), n ≥ 1} is bounded in L2 (Ω; H). Hence, by Lemma 1.2.3 ϕ(F ) ∈ D1,2 and {D(ϕn (F )), n ≥ 1} converges in the weak topology of L2 (Ω; H) to D(ϕ(F )). On the other hand, the sequence {∇ϕn (F ), n ≥ 1} is bounded by K. Hence, there exists a subsequence {∇ϕn(k) (F ), k ≥ 1} that converges to some random vector G = (G1 , . . . , Gm ) in the weak topology σ(L2 (Ω; Rm )). Moreover, G is bounded by K. Then, taking the limit in (1.36), we obtain Eq. (1.35). The proof of the lemma is now complete. If the law of the random vector F is absolutely continuous with respect to the Lebesgue measure on Rm , then Gi = ∂i ϕ(F ) in (1.35). Proposition 1.2.4 and Lemma 1.2.3 still hold if we replace D1,2 by D1,p for any p > 1. In fact, this follows from Lemma 1.5.3 and the duality relationship between D and δ.

30

1. Analysis on the Wiener space

We will make use of the following technical result. Lemma 1.2.4 The family of random variables {1, W (h)G − Dh G, G ∈ Sb , h ∈ H} is total in L2 (Ω). Proof: Fix h ∈ H, n, N ≥ 1, and set GN = W (h)n ψ N (W (h)), where ψ N is an inﬁnitely diﬀerentiable function such that 0 ≤ ψ N ≤ 1, ψ N (x) = 0 if |x| ≥ N + 1, ψ N (x) = 1 if |x| ≤ N , and supx,N |ψ N (x)| < ∞. Then, 2 W (h)GN − Dh GN converges in L2 (Ω) to W (h)n+1 − n hH W (h)n−1 as N tends to inﬁnity. Hence the closed linear span of the family contains all powers W (h)n , n ≥ 1, h ∈ H, which implies the result. Proposition 1.2.5 Let F be a random variable of the space D1,1 such that DF = 0. Then F = E(F ). Proof: If F ∈ D1,2 , then the result follows directly from Proposition 1.2.2. In the general case, let ψ N be a function in Cb∞ (R) such that ψ N (x) = 0 if |x| ≥ N + 1, ψ N (x) = x if |x| ≤ N . Let Fn be a sequence of smooth random Fn H ) tends to zero variables converging in L1 (Ω) to F and such that E(DF as n tends to inﬁnity. Then using Lemma 1.2.1 we obtain for any G ∈ Sb and any h ∈ H ! " E ψ N (F Fn ) W (h)G − Dh G

! " = E ψ N (F Fn )W (h)G − Dh (Gψ N (F Fn )) " ! Fn )) +E GDh (ψ N (F " ! Fn )) . = E GDh (ψ N (F

Taking the limit as n tends to inﬁnity yields ! " E ψ N (F ) W (h)G − Dh G = 0. As a consequence, by Lemma 1.2.4 E [ψ N (F )] = ψ N (F ) for each N . Hence, F = E(F ). Proposition 1.2.6 Let A ∈ F. Then the indicator function of A belongs to D1,1 if and only if P (A) is equal to zero or one. Proof: By the chain rule (Proposition 1.2.3) applied to to a function ϕ ∈ C0∞ (R), which is equal to x2 on [0, 1], we have D1A = D(1A )2 = 21A D1A and, therefore, D1A = 0 because from the above equality we get that this derivative is zero on Ac and equal to twice its value on A. So, by Proposition 1.2.5 we obtain 1A = P (A).

1.2 The derivative operator

31

Remarks: 1. If the underlying Hilbert space H is ﬁnite-dimensional, then the spaces Dk,p can be identiﬁed as ordinary Sobolev spaces of functions on Rn that together with their k ﬁrst partial derivatives have moments of order p with respect to the standard normal law. We refer to Ocone [270] for a detailed discussion of this fact. See also Exercise 1.2.8. 2. The above deﬁnitions can be exended to Hilbert-valued random variables. Let V be a real separable Hilbert space. Consider the family SV of V -valued smooth random variables of the form F =

n

F j vj ,

vj ∈ V,

Fj ∈ S.

j=1

n Deﬁne Dk F = j=1 Dk Fj ⊗ vj , k ≥ 1. Then Dk is a closable operator from SV ⊂ Lp (Ω; V ) into Lp (Ω; H ⊗k ⊗ V ) for any p ≥ 1. For any integer k ≥ 1 and any real number p ≥ 1 we can deﬁne the seminorm on SV ⎡ F k,p,V = ⎣E(F pV ) +

k

⎤ p1 E(Dj F pH ⊗j ⊗V )⎦ .

(1.37)

j=1

The operator Dk and the seminorms · k,p,V verify properties (i), (ii), and (iii) . We deﬁne the space Dk,p (V ) as the completion of SV with respect 1 to the norm · k,p,V . For k = 0 we put F 0,p,V = [E(F pV )] p , and D0,p (V ) = Lp (Ω; V ).

1.2.1 The derivative operator in the white noise case We will suppose in this subsection that the separable Hilbert space H is an L2 space of the form H = L2 (T, B, µ), where µ is a σ-ﬁnite atomless measure on a measurable space (T, B). The derivative of a random variable F ∈ D1,2 will be a stochastic process denoted by {Dt F, t ∈ T } due to the identiﬁcation between the Hilbert spaces L2 (Ω; H) and L2 (T × Ω). Notice that Dt F is deﬁned almost everywhere (a.e.) with respect to the measure µ × P . More generally, if k ≥ 2 and F ∈ Dk,2 , the derivative Dk F = {Dtk1 ,...,tk F, ti ∈ T }, is a measurable function on the product space T k × Ω, which is deﬁned a.e. with respect to the measure µk × P . Example 1.2.1 Consider the example of a d-dimensional Brownian motion on the interval [0, 1], deﬁned on the canonical space Ω = C0 ([0, 1]; Rd ).

32

1. Analysis on the Wiener space

In this case DF, hH can be interpreted as a directional Fr´chet ´ derivative. In fact, let us introduce the subspace H 1 of Ω which consists of all absolutely continuous functions x : [0, 1] → Rd with a square integrable t ˙ x˙ ∈ H = L2 ([0, 1]; Rd ). The space H 1 derivative, i.e., x(t) = 0 x(s)ds, is usually called the Cameron-Martin space. We can transport the Hilbert space structure of H to the space H 1 by putting d

x, yH 1 = x, ˙ y ˙ H=

i=1

1

x˙ i (s)y˙ i (s)ds.

0

In this way H 1 becomes a Hilbert space isomorphic to H. The injection of H 1 into Ω is continuous because we have 1 |x(s)|ds ˙ ≤ x ˙ H = xH 1 . sup |x(t)| ≤ 0≤t≤1

0

Assume d = 1 and consider a smooth functional of the particular form F = f (W (t1 ), . . . , W (tn )), f ∈ Cp∞ (Rn ), 0 ≤ t1 < · · · < tn ≤ 1, where t W (ti ) = 0 i dW Wt = W (1[0,ti ] ). Notice that such a functional is continuous in Ω. Then, for any function h in H, the scalar product DF, hH coincides · with the directional derivative of F in the direction of the element 0 h(s)ds, which belongs to H 1 . In fact, DF, hH

= =

n i=1 n

∂i f (W (t1 ), . . . , W (tn ))1[0,ti ] , hH

i=1

=

ti

∂i f (W (t1 ), . . . , W (tn ))

d F (ω + d

h(s)ds 0

·

h(s)ds)|=0 . 0

On the other hand, if F is Frechet ´ diﬀerentiable and λF denotes the signed measure associated with the Frechet ´ derivative of F , then Dt F = λF ((t, 1]). In fact, for any h ∈ H we have 1 t 1 F λ (dt)( h(s)ds)dt = λF ((t, 1])h(t)dt. DF, hH = 0

0

0

Suppose that F is a square integrable random variable having an orthogonal Wiener series of the form F =

∞

In (ffn ),

(1.38)

n=0

where the kernels fn are symmetric functions of L2 (T n ). The derivative Dt F can be easily computed using this expression.

1.2 The derivative operator

33

Proposition 1.2.7 Let F ∈ D1,2 be a square integrable random variable with a development of the form (1.38). Then we have Dt F =

∞

nIIn−1 (ffn (·, t)).

(1.39)

n=1

Proof: Suppose ﬁrst that F = Im (ffm ), where fm is a symmetric and elementary function of the form (1.10). Then Dt F =

m

m

ai1 ···im W (Ai1 ) · · · 1Aij (t) · · · W (Aim ) = mIIm−1 (ffm (·, t)).

j=1 1i1 ,...,im =1

Then the result follows easily. The heuristic meaning of the preceding proposition is clear. Suppose that F is a multiple stochastic integral of the form In (ffn ), which has also been denoted by ··· fn (t1 , . . . , tn )W (dt1 ) · · · W (dtn ). F = T

T

Then, F belongs to the domain of the derivation operator and Dt F is obtained simply by removing one of the stochastic integrals, letting the variable t be free, and multiplying by the factor n. Now we will compute the derivative of a conditional expectation with respect to a σ-ﬁeld generated by Gaussian stochastic integrals. Let A ∈ B. We will denote by FA the σ-ﬁeld (completed with respect to the probability P ) generated by the random variables {W (B), B ⊂ A, B ∈ B0 }. We need the following technical result: Lemma 1.2.5 Suppose that F is a square integrable random variable with the representation (1.38). Let A ∈ B. Then E(F |F FA ) =

∞

In (ffn 1⊗n A ).

(1.40)

n=0

Proof: It suﬃces to assume that F = In (ffn ), where fn is a function in En . Also, by linearity we can assume that the kernel fn is of the form 1B1 ×···×Bn , where B1 , . . . , Bn are mutually disjoint sets of ﬁnite measure. In this case we have E(F |F FA )

= E(W (B1 ) · · · W (Bm )|F FA ) n = E (W (Bi ∩ A) + W (Bi ∩ Ac )) | FA i=1

= In (1(B1 ∩A)×···×(Bn ∩A) ).

34

1. Analysis on the Wiener space

Proposition 1.2.8 Suppose that F belongs to D1,2 , and let A ∈ B. Then FA ) also belongs to the space D1,2 , and we the conditional expectation E(F |F have: FA )) = E(Dt F |F FA )1A (t) Dt (E(F |F a.e. in T × Ω. Proof:

By Lemma 1.2.5 and Proposition 1.2.7 we obtain

FA )) = Dt (E(F |F

∞

⊗(n−1)

nIIn−1 (ffn (·, t)1A

)1A (t) = E(Dt F |F FA )1A (t).

n=1

Corollary 1.2.1 Let A ∈ B and suppose that F ∈ D1,2 is FA -measurable. Then Dt F is zero almost everywhere in Ac × Ω. Given a measurable set A ∈ B, we can introduce the space DA,2 of random variables which are diﬀerentiable on A as the closure of S with respect to the seminorm 2 2 (Dt F ) µ(dt) . F A,2 = E(F 2 ) + E A

Exercises 1.2.1 Let W = {W (t), 0 ≤ t ≤ 1} be a one-dimensional Brownian motion. 1 Wt . Show Let h ∈ L2 ([0, 1]), and consider the stochastic integral F = 0 ht dW that F has a continuous modiﬁcation on C0 ([0, 1]) if and only if there exists a signed measure µ on (0, 1] such that h(t) = µ((t, 1]), for all t ∈ [0, 1], almost everywhere with respect to the Lebesgue measure. Hint: If h is given by a signed measure, the result is achieved through integrating by parts. For the converse implication, show ﬁrst that the continuous modiﬁcation of F must be linear, and then use the Riesz representation theorem of linear continuous functionals on C([0, 1]). For a more general treatment of this problem, refer to Nualart and Zakai [268]. 1.2.2 Show that the expression of the derivative given in Deﬁnition 1.2.1 does not depend on the particular representation of F as a smooth functional. 1.2.3 Show that the operator Dk is closable from S into Lp (Ω; H ⊗k ). Hint: Let {F FN , N ≥ 1} be a sequence of smooth functionals that converges to zero in Lp and such that Dk FN converges to some η in Lp (Ω; H ⊗k ). Iterating the integration-by-parts formula (1.31), show that E(η, h1 ⊗· · ·⊗ hk F ξ) = 0 for all h1 , . . . , hk ∈ H, F ∈ Sb , and k W (hi )2 . ξ = exp − i=1

1.2 The derivative operator

35

1.2.4 Let fn be a symmetric function in L2 ([0, 1]n ). Deduce the following expression for the derivative of F = In (ffn ): Dt F

= n!

n i=1

{t1 <···
fn (t1 , . . . , tn−1 , t)

Wtn−1 , × dW Wt1 · · · dW with the convention tn = 1. 1.2.5 Let F ∈ Dk,2 be given by the expansion F = that Dtk1 ,...,tk F =

∞

∞

fn ). n=0 In (f

Show

n(n − 1) · · · (n − k + 1)IIn−k (ffn (·, t1 , . . . , tk )),

n=k

and E(Dk F 2L2 (T k ) ) =

∞ n=k

n!2 ffn 2L2 (T n ) . (n − k)!

∞

1.2.6 Suppose that F = n=0 In (ffn ) is a random variable belonging to 1 E(Dn F ) for every n ≥ 0 (cf. the space D∞,2 = ∩k Dk,2 . Show that fn = n! Stroock [321]). 1.2.7 Let F = exp(W (h)− 12 T h2s µ(ds)), h ∈ L2 (T ). Compute the iterated derivatives of F and the kernels of its expansion into the Wiener chaos. 1.2.8 Let e1 , . . . , en be orthonormal elements in the Hilbert space H. Denote by Fn the σ-ﬁeld generated by the random variables W (e1 ), . . . , W (en ). Show that an Fn -measurable random variable F belongs to D1,2 if and only if there exists a function f in the weighted Sobolev space W1,2 (Rn , N (0, In )) such that F = f (W (e1 ), . . . , W (en )).

n Moreover, it holds that DF = i=1 ∂i f (W (e1 ), . . . , W (en ))ei . 1.2.9 Let (Ω, F, P ) be the canonical probability space of the standard Brownian motion on the time interval [0, 1]. Let F be a random variable that satisﬁes the following Lipschitz property: · hs ds) − F (ω)| ≤ chH a.s., h ∈ H = L2 ([0, 1]). |F (ω + 0

Show that F ∈ D1,2 and DF H ≤ c a.s. In [92] Enchev and Stroock proved the reciprocal implication. Hint: Suppose that F ∈ L2 (Ω) (the general case is treated by a truncation argument). Consider a complete orthonormal system {ei , i ≥ 1} in H. Fn ), where Fn is the σ-ﬁeld generated by the random Deﬁne Fn = E(F |F variables W (e1 ), . . . , W (en ). Show that Fn = fn (W (e1 ), . . . , W (en )), where

36

1. Analysis on the Wiener space

fn is a Lipschitz function with a Lipschitz constant bounded by c. Use Fn H ≤ c, a.s. Exercise 1.2.8 to prove that Fn belongs to D1,2 and DF Conclude using Lemma 1.2.3. 1.2.10 Show that the operator deﬁned in (1.33) is closable in Lp (Ω), for all p ≥ 1. 1.2.11 Suppose that W = {W (t), 0 ≤ t ≤ 1} is a standard one-dimensional Brownian motion. Show that the random variable M = sup0≤t≤1 W (t) belongs to the space D1,2 , and Dt M = 1[0,T ] (t), where T is the a.s. unique point where W attains its maximum. Hint: Approximate the supremum of W by the maximum on a ﬁnite set (see Section 2.1.4). F1 H 1.2.12 Let F1 and F2 be two elements of D1,2 such that F1 and DF are bounded. Show that F1 F2 ∈ D1,2 and D(F F1 F2 ) = F1 DF F2 + F2 DF F1 . 1.2.13 Show the following Leibnitz rule for the operator Dk : |I| k−|I| Dtk1 ,...,tk (F G) = DI (F )DI c (G), F, G ∈ S, I⊂{t1 ,...,tk }

where for any subset I of {t1 , . . . , tk }, |I| denotes the cardinality of I. 1.2.14 Show that the set S0 is dense in Dk,p for any k ≥ 1, p ≥ 1.

1.3 The divergence operator In this section we consider the divergence operator, deﬁned as the adjoint of the derivative operator. If the underlying Hilbert space H is an L2 space of the form L2 (T, B, µ), where µ is a σ-ﬁnite atomless measure, we will interpret the divergence operator as a stochastic integral and we will call it the Skorohod integral because in the Brownian motion case it coincides with the generalization of the Itˆo stochastic integral to anticipating integrands introduced by Skorohod [315]. We will deduce the expression of the Skorohod integral in terms of the Wiener chaos expansion as well as prove some of its basic properties. We will ﬁrst introduce the divergence operator in the framework of a Gaussian isonormal process W = {W (h), h ∈ H} associated with the Hilbert space H. We assume that W is deﬁned on a complete probability space (Ω, F, P ), and that F is generated by W . We recall that the derivative operator D is a closed and unbounded operator with values in L2 (Ω; H) deﬁned on the dense subset D1,2 of L2 (Ω). Deﬁnition 1.3.1 We denote by δ the adjoint of the operator D. That is, δ is an unbounded operator on L2 (Ω; H) with values in L2 (Ω) such that:

1.3 The divergence operator

37

(i) The domain of δ, denoted by Dom δ, is the set of H-valued square integrable random variables u ∈ L2 (Ω; H) such that |E(DF, uH )| ≤ cF 2 ,

(1.41)

for all F ∈ D1,2 , where c is some constant depending on u. (ii) If u belongs to Dom δ, then δ(u) is the element of L2 (Ω) characterized by (1.42) E(F δ(u)) = E(DF, uH ) for any F ∈ D1,2 . The operator δ is called the divergence operator and is closed as the adjoint of an unbounded and densely deﬁned operator. Let us study some basic properties of this operator.

1.3.1 Properties of the divergence operator Taking F = 1 in (1.42) we obtain E(δ(u)) = 0 if u ∈ Dom δ. Also, δ is a linear operator in Dom δ. We denote by SH the class of smooth elementary elements of the form n u= Fj h j , (1.43) j=1

where the Fj are smooth random variables, and the hj are elements of H. From the integration-by-parts formula established in Lemma 1.2.2 we deduce that an element u of this type belongs to the domain of δ and moreover that δ(u) =

n j=1

Fj W (hj ) −

n

DF Fj , hj H .

(1.44)

j=1

The following proposition provides a large class of H-valued random variables in the domain of the divergence. Note that if u ∈ D1,2 (H) then the derivative Du is a square integrable random variable with values in the Hilbert space H ⊗ H, which can be indentiﬁed with the space of HilbertSchmidt operators from H to H. Proposition 1.3.1 The space D1,2 (H) is included in the domain of δ. If u, v ∈ D1,2 (H), then E (δ(u)δ(v))) = E (u, vH ) + E (Tr (Du ◦ Dv)) .

(1.45)

In order to prove Proposition 1.3.1 we need the following commutativity relationship between the derivative and divergence operators. Let u ∈ SH , F ∈ S and h ∈ H. Then Dh (δ(u)) = u, hH + δ(Dh u).

(1.46)

38

1. Analysis on the Wiener space

Proof of (1.46): Suppose that u has the form (1.43). From (1.44) we deduce

Dh (δ(u))

=

n

Fj h, hj H +

j=1

n h $ # D Fj W (hj ) − D Dh Fj , hj H j=1

= u, hH + δ(D u). h

Notice that (1.46) is just a “Heisenberg commutativity relationship” that can be written, using commutator brackets, as [Dh , δ]u = u, hH . Proof of Proposition (1.3.1): Suppose ﬁrst that u, v ∈ SH . Let {ei , i ≥ 1} be a complete orthonormal system on H. Using the duality relationship (1.42) and property (1.46) we obtain E (δ(u)δ(v)))

= E (v, D(δ(u))H ) = E = E

∞ i=1

∞

v, ei H D (δ(u))

i=1

ei

v, ei H (u, ei H + δ(Dei u)) ⎛

= E (u, vH ) + E ⎝

∞

⎞ Dei u, ej H Dej v, ei H ⎠

i,j=1

= E (u, vH ) + E (Tr (Du ◦ Dv)) . As a consequence, we obtain the estimate 2 2 2 E δ(u)2 ≤ E uH + E DuH⊗H = u1,2,H .

(1.47)

This implies that the space D1,2 (H) is included in the domain of δ. In fact, if u ∈ D1,2 (H), there exists a sequence un ∈ SH such that un converges to u in L2 (Ω) and Dun converges to Du in L2 (Ω; H ⊗ H). Therefore, δ(un ) converges in L2 (Ω) and its limit is δ(u). Moreover, (1.45) holds for any u, v ∈ D1,2 (H). In order to extend the equality (1.46) to a more general class of random variables we need the following technical lemma. Lemma 1.3.1 Let G be a square integrable random variable. Suppose there exists Y ∈ L2 (Ω) such that E(Gδ(hF )) = E (Y F ) , for all F ∈ D1,2 . Then G ∈ Dh,2 and Dh G = Y .

1.3 The divergence operator

Proof:

39

We have

E (Y F ) = E(Gδ(hF )) =

∞

E ((J Jn G) δ(hF )) =

n=1

∞

E(F Dh (J Jn G)),

n=1

hence, Jn−1 Y = Dh (J Jn G) for each n ≥ 1 and this implies the result.

Proposition 1.3.2 Suppose that u ∈ D1,2 (H), and Dh u belongs to the domain of the divergence. Then δ(u) ∈ Dh,2 , and the commutation relation (1.46) holds. Proof: (1.42)

For all F ∈ D1,2 we have using (1.45) and the duality relationship E(δ(u)δ(hF ))

# $ = E u, hH F + Dh u, DF H = E u, hH + δ(Dh u) F ,

which implies the result, taking into account Lemma 1.3.1 The following proposition allows us to factor out a scalar random variable in a divergence. Proposition 1.3.3 Let F ∈ D1,2 and u be in the domain of δ such that F u ∈ L2 (Ω; H). Then F u belongs to the domain of δ and the following equality is true (1.48) δ(F u) = F δ(u) − DF, uH , provided the right-hand side of (1.48) is square integrable. Proof:

For any smooth random variable G ∈ S0 we have E (DG, F uH ) = E (u, D(F G) − GDF H ) = E ((δ(u)F − u, DF H ) G) ,

which implies the desired result. The next proposition is a version of Proposition 1.3.3, where u is replaced by a deterministic element h ∈ H. In this case it suﬃces to impose that F is diﬀerentiable in the direction of h (see Lemma 1.3.2 for a related result). Proposition 1.3.4 Let h ∈ H and F ∈ Dh,2 . Then F h belongs to the domain of δ and the following equality is true δ(F h) = F W (h) − Dh F. Proof: Suppose ﬁrst that F ∈ S. Then, the result is clearly true and using (1.45) yields 2 2 2 E (δ(F h)) = E F 2 hH + E Dh F . (1.49)

40

1. Analysis on the Wiener space

Finally, if Fn ∈ S is a sequence of smooth random variables converging to F in L2 (Ω) and such that Dh Fn converges to Dh F in L2 (Ω), then by (1.49) the sequence δ(F Fn h) is convergent in L2 (Ω). The following extension of Proposition 1.3.3 will be useful. Proposition 1.3.5 Suppose that H = L2 (T, B, µ). Let A ∈ B, and consider a random variable F ∈ DA,2 . Let u be an element of L2 (Ω; H) such that u1A belongs to the domain of δ and such that F u1A ∈ L2 (Ω; H). Then F u1A belongs to the domain of δ and the following equality is true δ(F u1A ) = F δ(u1A ) − Dt F ut µ(dt), (1.50) A

provided the right-hand side of (1.48) is square integrable. The next proposition provides a useful criterion to for the existence of the divergence. Proposition 1.3.6 Consider an element u ∈ L2 (Ω; H) such that there exists a sequence un ∈ Domδ which converges to u in L2 (Ω; H). Suppose that there exists G ∈ L2 (Ω) such that limn→∞ E(δ(un )F ) = E(GF ) for all F ∈ S. Then, u belongs to Domδ and δ(u) = G.

1.3.2 The Skorohod integral We will suppose in this subsection that the separable Hilbert space H is an L2 space of the form H = L2 (T, B, µ), where µ is a σ-ﬁnite atomless measure on a measurable space (T, B). In this case the elements of Domδ ⊂ L2 (T × Ω) are square integrable processes, and the divergence δ(u) is called the Skorohod stochastic integral of the process u. We will use the following notation: ut dW Wt . δ(u) = T

Any element u ∈ L2 (T × Ω) has a Wiener chaos expansion of the form u(t) =

∞

In (ffn (·, t)),

(1.51)

n=0

where for each n ≥ 1, fn ∈ L2 (T n+1 ) is a symmetric function in the ﬁrst n variables. Furthermore ∞ u(t)2 µ(dt) = n!ffn 2L2 (T n+1 ) . E T

n=0

The following result expresses the operator δ in terms of the Wiener chaos decomposition.

1.3 The divergence operator

41

Proposition 1.3.7 Let u ∈ L2 (T × Ω) with the expansion (1.51). Then u belongs to Dom δ if and only if the series δ(u) =

∞

In+1 (fn )

(1.52)

n=0

converges in L2 (Ω). Observe that the (n + 1)-dimensional kernels fn appearing in formula (1.51) are not symmetric functions of all its variables (only on the ﬁrst n variables). For this reason, the symmetrization of fn in all its variables will be given by fn (t1 , . . . , tn , t)

=

1 [ffn (t1 , . . . , tn , t) n+1 n + fn (t1 , . . . , ti−1 , t, ti+1 , . . . , tn , ti )]. i=1

Equation (1.52) can also be written without symmetrization, because for each n, In+1 (ffn ) = In+1 (fn ). However, the symmetrization is needed in order to compute the L2 norm of the stochastic integrals (see formula (1.53) ahead). Proof: Suppose that G = In (g) is a multiple stochastic integral of order n ≥ 1 where g is symmetric. Then we have the following equalities: E

ut Dt Gµ(dt)

=

T

∞ m=0

E (IIm (ffm (·, t))nIIn−1 (g(·, t))) µ(dt)

T

=

E (IIn−1 (ffn−1 (·, t))nIIn−1 (g(·, t))) µ(dt) = n(n − 1)! ffn−1 (·, t), g(·, t)L2 (T n−1 ) µ(dt) T

T

= n!ffn−1 , gL2 (T n ) = n!fn−1 , gL2 (T n ) = E In (fn−1 )IIn (g) = E In (fn−1 )G . Suppose ﬁrst that u ∈ Dom δ. Then from the above computations and from formula (1.42) we deduce that E(δ(u)G) = E(IIn (fn−1 )G) for every multiple stochastic integral G = In (g). This implies that In (fn−1 ) coincides with the projection of δ(u) on the nth Wiener chaos. Consequently, the series in (1.52) converges in L2 (Ω) and its sum is equal to δ(u).

42

1. Analysis on the Wiener space

Conversely, suppose that this series converges and let us denote its sum by V . Then from the preceding computations we have N N E ut Dt In (gn ) µ(dt) = E(V In (gn )) T

n=0

n=0

for all N ≥ 0. So we get |E( ut Dt F µ(dt))| ≤ V 2 F 2 , T

for any random variable F with a ﬁnite Wiener chaos expansion. By a density argument, this relation holds for any random variable F in D1,2 , and by Deﬁnition 1.3.1 we conclude that u belongs to Dom δ. From Proposition 1.3.7 it is clear that the class Dom δ of Skorohod integrable processes coincides with the subspace of L2 (T × Ω) formed by the processes that satisfy the following condition: 2

E(δ(u) ) =

∞

(n + 1)!fn 2L2 (T n+1 ) < ∞.

(1.53)

n=0

The space D1,2 (L2 (T )), denoted by L1,2 , coincides with the class of processes u ∈ L2 (T × Ω) such that u(t) ∈ D1,2 for almost all t, and there exists a measurable version of the two-parameter process Ds ut verifying E T T (Ds ut )2 µ(ds)µ(dt) < ∞. This space is included in Dom δ by Proposition 1.3.1. We recall that L1,2 is a Hilbert space with the norm u21,2,L2 (T ) = u2L2 (T ×Ω) + Du2L2 (T 2 ×Ω) . Note that L1,2 is isomorphic to L2 (T ; D1,2 ). If u and v are two processes in the space L1,2 , then Equation (1.45) can be written as E(δ(u)δ(v)) = E(ut vt )µ(dt) + E(Ds ut Dt vs )µ(ds)µ(dt). (1.54) T

T

T

Suppose that T = [0, ∞) and that µ is the Lebesgue measure. Then, if both processes are adapted to the ﬁltration generated by the Brownian motion, by Corollary 1.2.1 we have that Ds ut = 0 for almost all (s, t) such that s > t, since Ft = F[0,t] . Consequently, the second summand in Eq. (1.54) is equal to zero, and we recover the usual isometry property of the Ito ˆ integral. We could ask in which sense the Skorohod integral can be interpreted as an integral. Suppose that u is a smooth elementary process of the form u(t) =

n j=1

Fj hj (t),

(1.55)

1.3 The divergence operator

43

where the Fj are smooth random variables, and the hj are elements of L2 (T ). Equation (1.44) can be written as n n ut dW Wt = Fj hj (t)dW Wj − Dt Fj hj (t)µ(dt). (1.56) T

j=1

T

j=1

T

We see here that the Skorohod integral of a smooth elementary process can be decomposed into two parts, one that can be considered as a path-wise integral, and another that involves the derivative operator. We remark that if for every j the function hj is an indicator 1Aj of a set Aj ∈ B0 , and Fj is FAcj -measurable, then by Corollary 1.2.1, the second summand of Eq. (1.44) vanishes and the Skorohod integral of u is just the ﬁrst summand of (1.44). Proposition 1.3.2 can be reformulated as follows. Proposition 1.3.8 Suppose that u ∈ L1,2 . Assume that for almost all t and there is a version of the process {D t us , s ∈ T } is Skorohod integrable, Ws , t ∈ T } which is in L2 (T × Ω). Then δ(u) ∈ D1,2 , the process { T Dt us dW and we have Dt us dW Ws . (1.57) Dt (δ(u)) = ut + T

The next result characterizes the family of stochastic processes that can be written as DF for some random variable F . Proposition 1.3.9 Suppose that u ∈ L2 (T × Ω). There exists a random variable F ∈ D1,2 such that DF = u if and only if the kernels fn appearing in the integral decomposition (1.51) of u are symmetric functions of all the variables. Proof: The condition is obviously necessary. To show the suﬃciency, deﬁne F =

∞

1 In+1 (ffn ). n + 1 n=0

Clearly, this series converges in D1,2 and DF = u.

Proposition 1.3.10 Every process u ∈ L2 (T × Ω) has a unique orthogonal decomposition u = DF + u0 , where F ∈ D1,2 , E(F ) = 0, and E(DG, u0 H ) = 0 for all G in D1,2 . Furthermore, u0 is Skorohod integrable and δ(u0 ) = 0. Proof: The elements of the form DF , F ∈ D1,2 , constitute a closed subspace of L2 (T × Ω) by Proposition 1.3.9. Therefore, any process u ∈ L2 (T × Ω) has a unique orthogonal decomposition u = DF + u0 , where F ∈ D1,2 , and u0 ⊥DG for all G in D1,2 . From E(DG, u0 H ) = 0 for all G in D1,2 , it is clear that u0 is Skorohod integrable and δ(u0 ) = 0.

44

1. Analysis on the Wiener space

1.3.3 The Ito ˆ stochastic integral as a particular case of the Skorohod integral It is not diﬃcult to construct processes u that are Skorohod integrable (they belong to Dom δ) and do not belong to the space L1,2 . The next result provides a simple method for constructing processes of this type. Lemma 1.3.2 Let A belong to B0 , and let F be a square integrable random variable that is measurable with respect to the σ-ﬁeld FAc . Then the process F 1A is Skorohod integrable and δ(F 1A ) = F W (A). Proof: Suppose ﬁrst that F belongs to the space D1,2 . In that case using (1.48) and Corollary 1.2.1, we have Dt F 1A (t)µ(dt) = F W (A). δ(F 1A ) = F W (A) − T

Then, the general case follows by a limit argument, using the fact that δ is closed. Notice that Lemma 1.3.2 is a particular case of Proposition 1.3.4 because if F is in L2 (Ω, FAc , P ), then F ∈ D1A ,2 and D1A F = 0. Using this lemma we can show that the operator δ is an extension of the Itoˆ integral in the case of the Brownian motion. Let W = {W i (t); 0 ≤ t ≤ 1, 1 ≤ i ≤ d} be a d-dimensional Brownian motion. We denote by L2a the closed subspace of L2 ([0, 1] × Ω; Rd ) ∼ = L2 (T × Ω) (we recall that here T = [0, 1] × {1, . . . , d}) formed by the adapted processes. In this context we have the following proposition. Proposition 1.3.11 L2a ⊂ Dom δ, and the operator δ restricted to L2a coincides with the Ito ˆ integral, that is, δ(u) =

d i=1

Proof:

1

uit dW Wti .

0

Suppose that u is an elementary adapted process of the form ut =

n

Fj 1(tj ,tj+1 ] (t),

j=1

where Fj ∈ L2 (Ω, Ftj , P ; Rd ), and 0 ≤ t1 < · · · < tn+1 ≤ 1 (here Ft = F[0,t] ). Then from Lemma 1.3.2 we obtain u ∈ Dom δ and δ(u) =

d n i=1 j=1

Fji (W i (tj+1 ) − W i (tj )).

(1.58)

1.3 The divergence operator

45

We know that any process u ∈ L2a can be approximated in the norm of L2 (T × Ω) by a sequence un of elementary adapted processes. Then by (1.58) δ(un ) is equal to the Itˆˆo integral of un and it converges in L2 (Ω) to the Ito ˆ integral of u. Since δ is closed we deduce that u ∈ Dom δ, and δ(u) is equal to the Itoˆ integral of u. More generally, any type of adapted stochastic integral with respect to a multiparameter Gaussian white noise W can be considered as a Skorohod integral (see, for instance, Nualart and Zakai [264]). Let W = {W (t), t ∈ [0, 1]} be a one-dimensional Brownian motion. We are going to introduce a class of processes which are diﬀerentiable in the future and it contains L2a . The Skorohod integral is well deﬁned in this class and possesses properties similar to those of the Itˆo integral (see Chapter 3). Let L1,2,f be the closure of SH with respect to the seminorm 1 2 2 2 u1,2,f = E ut dt + E (Ds ut ) dsdt , s≤t

0

and let L be deﬁned as the closure of SH with respect to the seminorm 2 2 uF = u1,2,f + E (Dr Ds ut )2 dsdt . F

r ∨s≤t

Remarks: 1. LF coincides with the class of processes u ∈ L1,2,f such that {Ds ut 1[0,s] (t), t ∈ [0, 1]} belongs to D1,2 (L2 ([0, 1]2 )). b 2. If u ∈ L1,2,f , then a ut dt ∈ D1[b,1] ,2 for any 0 ≤ a < b ≤ 1. Proposition 1.3.12 L2a ⊂ LF , and for any u ∈ L2a we have Ds ut = 0 if t ≥ s, and 1 2 2 uF = E ut dt . (1.59) 0

Proof: Let u be an elementary adapted process of the form (1.58). Then u ∈ LF , and Ds ut = 0 if t ≥ s. The result follows because these processes are dense in L2a . Proposition 1.3.13 LF ⊂ Domδ and for all u ∈ LF we have 2 E δ(u)2 ≤ 2 uF . Proof:

If u ∈ SH , then by (1.54) we have 1 2 2 E δ(u) = E ut dt + E 0

= E

0

1

Ds ut Dt us dsdt

0

1

1

u2t dt + 2E 0

0

1

0

t

Ds ut Dt us dsdt . (1.60)

46

1. Analysis on the Wiener space

Using the duality between the operators δ and D and applying again (1.54) yields 1 t 1 t Ds ut Dt us dsdt = E ut Dt us dW Ws dt E 0

0

0

0

1 E u2t dt 2 0 2 1 t 1 + E Dt us dW Ws dt . (1.61) 2 0 0 ≤

1

Moreover, (1.54) yields 2 1 t Dt us dW Ws dt = E E 0

0

1

t

t

+E

0

2

(Dr Dt us ) drdsdt . 0

0

1

t

2

(Dt us ) dsdt 0

(1.62)

0

Substituting (1.61) and (1.62) into (1.60) we obtain the inequality (1.59). Finally, the general case follow by a density argument.

1.3.4 Stochastic integral representation of Wiener functionals Suppose that W = {W (t), t ∈ [0, 1]} is a one-dimensional Brownian motion. We have seen in Section 1.1.3 that any square integrable random variable F , measurable with respect to W , can be written as 1 φ(t)dW Wt , F = E(F ) + 0

L2a .

where the process φ belongs to When the variable F belongs to the space D1,2 , it turns out that the process φ can be identiﬁed as the optional projection of the derivative of F . This is called the Clark-Ocone representation formula: Proposition 1.3.14 Let F ∈ D1,2 , and suppose that W is a one-dimensional Brownian motion. Then 1 F = E(F ) + E(Dt F |F Ft )dW Wt . (1.63) Proof:

Suppose that F =

E(Dt F |F Ft )

= =

∞ n=1 ∞ n=1

∞

0

fn ). n=0 In (f

Using (1.39) and (1.40) we deduce

nE(IIn−1 (ffn (·, t))|F Ft ) nIIn−1 fn (t1 , . . . , tn−1 , t)1{t1 ∨···∨tn−1 ≤t} .

1.3 The divergence operator

47

Set φt = E(Dt F |F Ft ). We can compute δ(φ) using the above expression for φ and (1.52), and we obtain δ(φ) =

∞

In (ffn ) = F − E(F ),

n=1

which shows the desired result because δ(φ) is equal to the Itˆo stochastic integral of φ. As a consequence of this integral representation, and applying the H¨ older, Burkholder, and Jensen inequalities, we deduce the following inequality for F ∈ D1,p and p ≥ 2 in the case T = [0, 1]: E(|F | ) ≤ Cp [|E(F )| + E( p

p

1

|Dt F |p dt)].

0

1.3.5 Local properties In this subsection we will show that the divergence and derivative operators verify a local property. The local property of the Skorohod integral is analogous to that of the Itˆ o integral. Proposition 1.3.15 Let u ∈ D1,2 (H) and A ∈ F, such that u = 0 on A. Then δ(u) = 0 a.s. on A. Proof:

Let F be a smooth random variable of the form F = f (W (h1 ), . . . , W (hn ))

with f ∈ C0∞ (Rn ). We want to show that δ(u)1{ u H =0} = 0 a.s. Suppose that φ : R → R is an inﬁnitely diﬀerentiable function such that φ ≥ 0, φ(0) = 1 and its support is included in the interval [−1, 1]. Deﬁne the function φ (x) = φ( x ) for all > 0. We will use (see Exercise 1.3.3) the fact that the product F φ (u2H ) belongs to D1,2 . Then by the duality relation (1.42) we obtain # $ E δ(u)φ u2H F = E u, D[F φ (u2H )] H = E φ u2H u, DF H + 2E F φ u2H u, Du uH . We claim that the above expression converges to zero as tends to zero. In fact, ﬁrst observe that the random variables V = φ u2H u, DF H + 2F φ u2H u, Du uH

48

1. Analysis on the Wiener space

converge a.s. to zero as ↓ 0, since uH = 0 implies V = 0. Second, we can apply the Lebesgue dominated convergence theorem because we have φ u2H u, DF ≤ φ∞ uH DF H , H φ u2H u, Du u H

≤ sup |xφ (x)| DuH⊗H ≤ φ ∞ DuH⊗H . x

The proof is now complete. Notice that the local property of the divergence has been established for H-valued random variables in the space D1,2 (H). We do not know if it holds for an arbitrary variable u in the domain of δ, although we know that in the Brownian case the local property holds in the subspace L2a , because as we have seen δ coincides there with the Itˆo integral. As an extension of this result we will prove in Proposition 1.3.17 below that the local property of δ holds in the space LF . The next result shows that the operator D is local in the space D1,1 . Proposition 1.3.16 Let F be a random variable in the space D1,1 such that F = 0 a.s. on some set A ∈ F. Then DF = 0 a.s. on A. Proof: We can assume that F ∈ D1,1 ∩ L∞ (Ω), replacing F by arctan(F ). We want to show that 1{F =0} DF = 0 a.s. Consider a function φ : R → R such as that in the proof of Proposition 1.3.15. Set x φ (y)dy. ψ (x) = −∞

By the chain rule ψ (F ) belongs to D1,1 and Dψ (F ) = φ (F )DF . Let u be a smooth elementary process of the form u=

n

Fj h j ,

j=1

where Fj ∈ Sb and hj ∈ H. Observe that the duality relation (1.42) holds for F in D1,1 ∩ L∞ (Ω) and for a process u of this type. Note that the class of such processes u is total in L1 (Ω; H) in the sense that if v ∈ L1 (Ω; H) satisﬁes E(v, uH ) = 0 for all u in the class, then v ≡ 0. Then we have |E (φ (F ) DF, uH )| = |E (D (ψ (F )) , uH )| = |E (ψ (F )δ(u))| ≤ φ∞ E(|δ(u)|). Letting ↓ 0, we obtain

E 1{F =0} DF, uH = 0,

which implies the desired result.

1.3 The divergence operator

49

Proposition 1.3.17 Suppose that W = {W (t), t ∈ [0, 1]} is a onedimensional Brownian motion. Let u ∈ LF and A ∈ F, such that ut (ω) = 0 a.e. on the product space [0, T ]× A. Then δ(u) = 0 a.s. on A. n deﬁned in Proof: Let u ∈ LF . Consider the sequence of processes u n has (1.17). As in Lemma 1.1.3 the operator Pn deﬁned by Pn (u) = u F F n u ) converges norm bounded by 1 from L to L . By Proposition 1.3.13 δ( in L2 (Ω) to δ(u) as n tends to inﬁnity. On the other hand, applying Proposition 1.3.4 we have −n n 2 −1 i2 n n 2 u(s)ds W(i+1)2−n − Wi2−n δ( u ) = (i−1)2−n

i=1

−

(i+1)2−n i2−n

i2−n

(i−1)2−n

Ds ut dtds.

and by the local property of the operator D in the space L1,2,f (see Exercise 1 1.3.12) we deduce that this expression is zero on the set 0 u2t dt = 0, which completes the proof of the proposition. We can localize the domains of the operators D and δ as follows. If L is a class of random variables (or processes) we denote by Lloc the set of random variables F such that there exists a sequence {(Ωn , Fn ), n ≥ 1} ⊂ F × L with the following properties: (i) Ωn ↑ Ω , a.s. (ii) F = Fn a.s. on Ωn . 1,p , then DF is deﬁned If F ∈ D1,p loc , p ≥ 1, and (Ωn , Fn ) localizes F in D without ambiguity by DF = DF Fn on Ωn , n ≥ 1. More generally, the iterated derivative Dk is well deﬁned by localization in the space Dk,p loc . Moreover, for any h ∈ H the operator Dh is also local (see Exercise 1.3.12) and it has a local domain Dh,p loc , p ≥ 1. 1,2 Then, if u ∈ Dloc (H), the divergence δ(u) is deﬁned as a random variable determined by the conditions

δ(u)|Ωn = δ(un )|Ωn

for

all

n ≥ 1,

where (Ωn , un ) is a localizing sequence for u. Although the local property of the divergence operator has not been proved in its domain, we can localize the divergence as follows. Suppose that {(Ωn , un ), n ≥ 1} is a localizing sequence for u in (Domδ)loc . If δ(un ) = δ(um ) a.s. on Ωn for all m ≥ n, then, the divergence δ(u) is the random variable determined by the conditions δ(u)|Ωn = δ(un )|Ωn for all n ≥ 1, but it may depend on the localizing sequence.

50

1. Analysis on the Wiener space

Examples: Let W = {W (t), t ∈ [0, 1]} be a one-dimensional Brownian motion. The processes ut =

Wt , |W W1 |

vt = exp(W Wt4 )

1,2 belong to Lloc . In fact, the sequence (Ωn , un ) with Ωn = {|W W1 | > n1 } and Wt n 1,2 ut = |W1 |∨(1/n) localizes the process u in L . On the other hand, if we take Wt | < n} and vtn = exp(W Wt4 ∧ n), we obtain a localizing Ωn = {supt∈[0,1] |W sequence for the process v (see also Exercise 1.3.10).

The following proposition asserts that the Skorohod integral deﬁned by localization in the space LF loc thanks to Proposition 1.3.17 is an extension of the Itˆ o stochastic integral. Proposition 1.3.18 Let W = {W Wt , t ∈ [0, 1]} be a one-dimensional Brown1 ian motion and consider an adapted process u such that 0 u2t dt < ∞ a. s. ˆ stochastic integral Then, u belongs to LF loc and δ(u) coincides with the Ito 1 u dW W . t 0 t Proof: For any integer k ≥ 1 consider an inﬁnitely diﬀerentiable function ϕk : R → R such that ϕk (x) = 1 if |x| ≤ k, ϕk (x) = 0 if |x| ≥ k + 1 and |ϕk (x)| ≤ 1 for all x. Deﬁne t k 2 us ds u t = u t ϕk 0

and

Ωk =

1

u2s ds ≤ k .

0

Then we have Ωk ↑ Ω a.s., u = uk on [0, 1] × Ωk , and uk ∈ L2a because uk is adapted and 1 1 t k 2 2 2 ut dt = u t ϕk us ds dt ≤ k + 1. 0

0

0

Then, the result follows because on L2a the Skorohod integral is an extension of the Itˆ oˆ integral. The following lemma is helpful in the application of the analysis on the Wiener space. It allows us to transform measurability properties with respect to σ-ﬁelds generated by variables of the ﬁrst chaos into analytical conditions. 1,2 . Given a closed subLemma 1.3.3 Let G be a random variable in Dloc space K of H, we denote by FK the σ-ﬁeld generated by the Gaussian random variables {W (h), h ∈ K}. Let A ∈ FK . Suppose that 1A G is FK measurable. Then DG ∈ K, a.s., in A.

1.3 The divergence operator

51

Proof: Since we can approximate G by ϕn (G), where ϕn ∈ Cb∞ (R), 1,2 ϕn (x) = x for |x| ≤ n, it is suﬃcient to prove the result for G ∈ Dloc ∩L2 (Ω). Let h ∈ H be an element orthogonal to K. Then E(G | FK ) belongs to Dh,2 and Dh E(G | FK ) = 0. However, G ∈ Dh,2 loc and G = E(G | FK ) a.s. on A. From the local property of Dh it follows that Dh G = 0 a.s. on A. Then it remains to choose a countable and dense set of elements h in the orthogonal complement of K, and we obtain that DG ∈ K a.s. on A. The following lemma shows that an Itˆ ˆo integral is diﬀerentiable if and only if its integrand is diﬀerentiable (see [279]). Lemma 1.3.4 Let W = {W (t), t ∈ [0, 1]} be a one-dimensional Brownian motion. Consider a square integrable adapted process u = {ut , t ∈ [0, 1]}, t and set Xt = 0 us dW Ws . Then the process u belongs to the space L1,2 if and only if X1 belongs to D1,2 . In this case the process X belongs to L1,2 , and we have t t t s 2 2 E(|Ds Xt | )ds = E(us )ds + E(|Dr us |2 )drds, (1.64) 0

0

0

0

for all t ∈ [0, 1]. Proof: Suppose ﬁrst that u ∈ L1,2 . Then the process u veriﬁes the hypothesis of Proposition 1.3.8 of the Skorohod integral. In fact, the process {Dt us , s ∈ [t, 1]} is Skorohod integrable because it is adapted and square integrable. Moreover, 2 1 1 1 1 Dt us dW Ws dt = E(|Dt us |2 )dsdt < ∞, E 0

0

t

t

due to the isometry of the Ito ˆ integral. Consequently, by Proposition 1.3.8 we obtain that Xt belongs to D1,2 for any t and t Ds Xt = us 1{s≤t} + Ds ur dW Wr . (1.65) s

Taking the expectation of the square of the above expression, we get (1.64) and X belongs to L1,2 . Conversely, suppose that X1 belongs to D1,2 . For each N wedenote by t N Ws . ut the projection of ut on PN = H0 ⊕ · · · ⊕ HN . Set XtN = 0 uN s dW N N Then Xt is the projection of Xt on PN +1 . Hence, X1 converges to X1 in the topology of the space D1,2 . Then the result follows from the inequality 1 1 1 s N 2 N 2 2 E(|Ds X1 | )ds = E(|us | )ds + E(|Dr uN s | )drds 0

0

≥

1

0

0

s

2 E(|Dr uN s | )drds = E DuN 2L2 ([0,1]2 ) . 0

0

52

1. Analysis on the Wiener space

Exercises 1.3.1 Show the isometry property (1.54) using the Wiener series expansion of the process u. 1.3.2 Let R be the class of processes of the form u=

n

Fi 1Ai ,

i=1

where Ai ∈ B0 , and Fi ∈ L2 (Ω, FAci , P ). Show that Dom δ coincides with the closed hull of R for the norm uL2 (T ×Ω) + δ(u)2 . 1.3.3 Let F be a smooth random variable of the form F = f (W (h1 ), . . . , W (hn )), where f ∈ C0∞ (Rn ). Let g be in C0∞ (R), and let u ∈ L1,2 . Show that F g(u2H ) belongs to the space D1,2 and D F g(u2H ) = DF g(u2H ) + 2F g (u2H )Duu . 1.3.4 Let F ∈ D1,2 be a random variable such that E(|F |−2 ) < ∞. Then P {F > 0} is zero or one. Hint: Using the duality relation, compute E(ϕ (F )δ(u)), where u is an arbitrary bounded element in the domain of δ and ϕ is an approximation of the sign function. 1.3.5 Show that the random variable F = 1{W (h)>0} does not belong to 1,2 , and DF = 0. D1,2 . Prove that it belongs to Dloc 1.3.6 Show the following diﬀerentiation rule (see Ocone and Pardoux [272, Lemma 2.3]) . Let F = (F 1 , . . . , F k ) be a random vector whose components 1,2 . Consider a measurable process u = {u(x), x ∈ Rk } which belong to Dloc can be localized by processes with continuously diﬀerentiable paths, such 1,2 and the derivative Du(x) has a continuous that for any x ∈ Rk , u(x) ∈ Dloc version as an H-valued process. Suppose that for any a > 0 we have ! " 2 2 < ∞, E sup |u(x)| + Du(x)H |x|≤a

sup |∇u(x)|∞ |x|≤a

< ∞.

1,2 , and we have Then the composition G = u(F ) belongs to Dloc

DG =

k i=1

∂i u(F )DF i + (Du)(F ).

1.3 The divergence operator

53

Hint: Approximate the composition u(F ) by the integral u(x)ψ (F − x)dx, Rk

where ψ is an approximation of the identity. 1.3.7 Suppose that H = L2 (T ). Let δ k be the adjoint of the operator Dk . That is, a multiparameter process u ∈ L2 (T k × Ω) belongs to the domain of δ k if and only if there exists a random variable δ k (u) such that E(F δ k (u)) = E(u, Dk F L2 (T k ) ) for all F ∈ Dk,2 . Show that a process u ∈ L2 (T k × Ω) with an expansion ut =

∞

In (ffn (·, t)),

t ∈ T k,

n=0

belongs to the domain of δ k if and only if the series δ k (u) =

∞

In+k (ffn )

n=0

converges in L2 (Ω). 1.3.8 Let u ∈ L2 (T k × Ω). Show that there exists a random variable F ∈

∞ fn (·, t)) and the Dk,2 such that u = Dk F if and only if ut = n=0 In (f kernels fn ∈ L2 (T n+k ) are symmetric functions of all their variables. Show that every process u ∈ L2 (T k × Ω) admits a unique decomposition u = Dk F + u0 , where F ∈ Dk,2 and δ k (u0 ) = 0. 1.3.9 Let {W Wt , t ∈ [0, 1]} be a one-dimensional Brownian motion. Using Exercise 1.2.6 ﬁnd the Wiener chaos expansion of the random variables F1 = 0

1

(t3 Wt3 + 2tW Wt2 )dW Wt ,

1

F2 =

teWt dW Wt .

0

1.3.10 Suppose that H = L2 (T ). Let u ∈ L1,2 and F ∈ D1,2 be two elements such that P (F = 0) = 0 and E T |ut Ds F |2 µ(ds)µ(dt) < ∞. 1,2 ut Show that the process |F | belongs to Lloc , and compute its derivative and its Skorohod integral. 1.3.11 In the particular case H = L2 (T ), deduce the estimate (1.47) from Equation (1.53) and the inequality f˜n L2 (T n+1 ) ≤ ffn L2 (T n+1 ) .

54

1. Analysis on the Wiener space

1.3.12 Let F ∈ Dh,p , p ≥ 1, be such that F = 0 a.s. on A ∈ F. Show that Dh F = 0 a.s. on A. As a consequence, deduce the local property of the operator D on the space L1,2,f . 1.3.13 Using Clark-Ocone formula (1.63) ﬁnd the stochastic integral representation of the following random variables: (i) F = W13 , (ii) F = exp(2W W1 ), (iii) F = sup0≤t≤1 Wt .

1.4 The Ornstein-Uhlenbeck semigroup In this section we describe the main properties of the Ornstein-Uhlenbeck semigroup and, in particular, we show the hypercontractivity property.

1.4.1 The semigroup of Ornstein-Uhlenbeck We assume that W = {W (h), h ∈ H} is an isonormal Gaussian process associated to the Hilbert space H deﬁned in a complete probability space (Ω, F, P ), and that F is generated by W . We recall that Jn denotes the orthogonal projection on the nth Wiener chaos. Deﬁnition 1.4.1 The Ornstein-Uhlenbeck semigroup is the one-parameter semigroup {T Tt , t ≥ 0} of contraction operators on L2 (Ω) deﬁned by Tt (F ) =

∞

e−nt Jn F,

(1.66)

n=0

for any F ∈ L2 (Ω) . There is an alternative procedure for introducing this semigroup. Suppose that the process W = {W (h), h ∈ H} is an independent copy of W . We will assume that W and W are deﬁned on the product probability space (Ω × Ω , F ⊗ F , P × P ). For any t > 0 we consider the process Z = {Z(h), h ∈ H} deﬁned by % Z(h) = e−t W (h) + 1 − e−2t W (h), h ∈ H. This process is Gaussian, with zero mean and with the same covariance function as W . In fact, we have E(Z(h1 )Z(h2 )) = e−2t h1 , h2 H + (1 − e−2t )h1 , h2 H = h1 , h2 H . Let W : Ω → RH and W : Ω → RH be the canonical mappings associated with the processes {W (h), h ∈ H} and {W (h), h ∈ H}, respectively. Given

1.4 The Ornstein-Uhlenbeck semigroup

55

a random variable F ∈ L2 (Ω), we can write F = ψ F ◦W , where ψ F is a measurable mapping from RH to R, determined P ◦W −1 a.s. √ As a consequence, the random variable ψ F (Z(ω, ω )) = ψ F (e−t W (ω) + 1 − e−2t W (ω )) is well deﬁned P × P a.s. Then, for any t > 0 we put % Tt (F ) = E (ψ F (e−t W + 1 − e−2t W )), (1.67) where E denotes mathematical expectation with respect to the probability P . Equation (1.67) is called Mehler’s formula. We are going to check the equivalence between (1.66) and (1.67). First we will see that both deﬁnitions give rise to a linear contraction operator on L2 (Ω). This is clear for the deﬁnition (1.66). On the other hand, (1.67) deﬁnes a linear contraction operator on Lp (Ω) for any p ≥ 1 because we have % E(|T Tt (F )|p ) = E(|E (ψ F (e−t W + 1 − e−2t W ))|p ) % ≤ E(E (|ψ F (e−t W + 1 − e−2t W )|p )) = E(|F |p ). So, to show that (1.66) is equal to (1.67) on L2 (Ω), it suﬃces to check that both deﬁnitions coincide when F = exp W (h) − 12 h2H , h ∈ H. We have % 1 −t 2 −2t E exp e W (h) + 1 − e W (h) − hH 2 ∞ W (h) 1 = exp e−t W (h) − e−2t h2H = e−nt hnH Hn 2 hH n=0 =

∞ e−nt In (h⊗n ). n! n=0

On the other hand, Tt (F )

= Tt =

∞ 1 ⊗n In (h ) n! n=0

∞ e−nt In (h⊗n ), n! n=0

which yields the desired equality. The operators Tt verify the following properties: (i) Tt is nonnegative (i.e., F ≥ 0 implies Tt (F ) ≥ 0). (ii) Tt is symmetric: E(GT Tt (F )) = E(F Tt (G)) =

∞ n=0

e−nt E(J Jn (F )J Jn (G)).

56

1. Analysis on the Wiener space

Example 1.4.1 The classical Ornstein-Uhlenbeck (O.U.) process on the real line {Xt , t ∈ R} is deﬁned as a Gaussian process with zero mean and covariance function given by K(s, t) = βe−α|s−t| , where α, β > 0 and s, t ∈ R. This process is Markovian and stationary, and these properties characterize the form of the covariance function, assuming that K is continuous. It is easy to check that the transition probabilities of the Ornstein-Uhlenbeck process Xt are the normal distributions P (Xt ∈ dy|Xs = x) = N (xe−α(t−s) , β(1 − e−2α(t−s) )). In fact, for all s < t we have = e−α(t−s) Xs , = β(1 − e−2α(t−s) ).

E(Xt |Xs ) E((Xt − E(Xt |Xs ))2 )

Also, the standard normal law ν = N (0, β) is an invariant measure for the Markov semigroup associated with the O.U. process. Consider the semigroup of operators on L2 (R, B(R), ν) determined by the stationary transition probabilities of the O.U. process (with α, β = 1). This semigroup is a particular case of the Ornstein-Uhlenbeck semigroup introduced in Deﬁnition 1.4.1, if we take (Ω, F, P ) = (R, B(R), ν), H = R, and W (t)(x) = tx for any t ∈ R. In fact, if {Xs , s ∈ R} is a real-valued O.U. process, for any bounded measurable function f on R we have for t ≥ 0 and s ∈ R f (y)P (Xs+t ∈ dy|Xs = x) = f (y)N (e−t x, 1 − e−2t )(dy) R R % f (e−t x + 1 − e−2t y)ν(dy) = R

=

(T Tt f )(x).

Let W be a Brownian measure on the real line. That is, {W (B), B ∈ B(R)} is a centered Gaussian family such that 1B1 ∩B2 (x)dx. E(W (B1 )W (B2 )) = R

Then the process Xt =

%

t

2αβ −∞

e−α(t−u) dW Wu

has the law of an Ornstein-Uhlenbeck process of parameters α, β. Furthermore, the process Xt satisﬁes the stochastic diﬀerential equation % Wt − αXt dt. dXt = 2αβdW

1.4 The Ornstein-Uhlenbeck semigroup

57

Consider now the case where H = L2 (T, B, µ) and µ is a σ-ﬁnite atomless measure. Using the above ideas we are going to introduce an OrnsteinUhlenbeck process parametrized by H. To do this we consider a Brownian F, P) and with measure B on T × R, deﬁned on some probability space (Ω, intensity equal to 2µ(dt)dx. Then we deﬁne t h(τ )e−(t−s) B(dτ , ds). (1.68) Xt (h) = −∞

T

It is easy to check that Xt (h) is a Gaussian zero-mean process with covariance function given by t (h1 )Xt (h2 )) = e−|t1 −t2 | h1 , h2 H . E(X 1 2 Consequently, we have the following properties: (i) For any h ∈ H, {Xt (h), t ∈ R} is a real-valued Ornstein-Uhlenbeck process with parameters α = 1 and β = h2H . (ii) For any t ≥ 0, {Xt (h), h ∈ H} has the same law as {W (h), h ∈ H}. Therefore, for any random variable F ∈ L0 (Ω) we can deﬁne the composition F (Xt ). That is, F (Xt ) is short notation for ψ F (Xt ), where ψ F is the mapping from RH to R determined by ψ F (W ) = F . Let Ft denote the σ-ﬁeld generated by the random variables B(G), where G is a measurable and bounded subset of T × (−∞, t]. The following result establishes the relationship between the process Xt (h) and the Ornstein-Uhlenbeck semigroup. Proposition 1.4.1 For any t ≥ 0, s ∈ R, and for any integrable random variable F we have (Xs+t )|Fs ) = (T Tt F )(Xs ). E(F

(1.69)

Proof: Without loss of generality we may assume that F is a smooth random variable of the form F = f (W (h1 ), . . . , W (hn )), where f ∈ Cp∞ (Rn ), h1 , . . . , hn ∈ H, 1 ≤ i ≤ n. In fact, the set S of smooth random variables is dense in L1 (Ω), and both members of Eq. (1.69) are continuous in L1 (Ω). We are going to use the decomposition Xs+t = Xs+t − e−t Xs + e−t Xs . Note that (i) {e−t Xs (h), h ∈ H} is Fs -measurable, and (ii) the Gaussian family {Xs+t (h) − e−t Xs (h), h ∈ H} has the same law √ −2t as { 1 − e W (h), h ∈ H}, and is independent of Fs .

58

1. Analysis on the Wiener space

Therefore, we have (Xs+t )|F˜s ) E(F

(Xs+t (h1 ), . . . , Xs+t (hn ))|Fs ) = E(f f (Xs+t (h1 ) − e−t Xs (h1 ) + e−t Xs (h1 )), = E . . . , Xs+t (hn ) − e−t Xs (hn ) + e−t Xs (hn ))|Fs % = E f 1 − e−2t W (h1 ) + e−t Xs (h1 ) , ..., =

%

1−

e−2t W (h

n ) + e Xs (hn )

−t

(T Tt F )(Xs ),

where W is an independent copy of W , and E denotes the mathematical expectation with respect to W . Consider, in particular, the case of the Brownian motion. That means Ω = C0 ([0, 1]), and P is the Wiener measure. In that case, T = [0, 1], and the process deﬁned by (1.68) can be written as 1 h(τ )Xt (dτ ), Xt (h) = t

0

τ

where Xt (τ ) = −∞ 0 e−(t−s) W (dσ, ds), and W is a two-parameter Wiener process on [0, 1] × R with intensity 2dtdx. We remark that the stochastic process {Xt (·), t ∈ R} is a stationary Gaussian continuous Markov process with values on C0 ([0, 1]), which has the Wiener measure as invariant measure.

1.4.2 The generator of the Ornstein-Uhlenbeck semigroup In this section we will study the properties of the inﬁnitesimal generator of the Ornstein-Uhlenbeck semigroup. Let F ∈ L2 (Ω) be a square integrable random variable. We deﬁne the operator L as follows: LF =

∞

−nJ Jn F,

n=0

provided this series converges in L2 (Ω). The domain of this operator will be the set Dom L = {F ∈ L2 (Ω), F =

∞ n=0

In (ffn ) :

∞

n2 J Jn F 22 < ∞}.

n=1

In particular, Dom L ⊂ D . Note that L is an unbounded symmetric operator on L2 (Ω). That is, E(F LG) = E(GLF ) for all F, G ∈ Dom L. The 1,2

1.4 The Ornstein-Uhlenbeck semigroup

59

next proposition tells us that L coincides with the inﬁnitesimal generator of the Ornstein-Uhlenbeck semigroup {T Tt , t ≥ 0} introduced in Deﬁnition 1.4.1. In particular, L is self-adjoint and (hence) closed. Proposition 1.4.2 The operator L coincides with the inﬁnitesimal generator of the Ornstein-Uhlenbeck semigroup {T Tt , t ≥ 0}. Proof: We have to show that F belongs to the domain of L if and only if the limit limt↓0 1t (T Tt F − F ) exists in L2 (Ω) and, in this case, this limit is equal to LF . Assume ﬁrst that F ∈ Dom L. Then 2 '2 ∞ & 1 1 −nt Tt F − F ) − LF (e E (T = − 1) + n E(|J Jn F |2 ), t t n=0 which converges to zero as t ↓ 0. In fact, for any n the expression 1t (e−nt − 1) + n tends to zero, and moreover | 1t (e−nt − 1)| ≤ n. Tt F − F ) = G in L2 (Ω). Then we have Conversely, suppose that limt↓0 1t (T that 1 Tt Jn F − Jn F ) = −nJ Jn G = lim (T Jn F. t↓0 t Therefore, F belongs to the domain of L, and LF = G.

The next proposition explains the relationship between the operators D, δ, and L. Proposition 1.4.3 δDF = −LF , that is, for F ∈ L2 (Ω) the statement F ∈ Dom L is equivalent to F ∈ Dom δD (i.e., F ∈ D1,2 and DF ∈ Dom δ), and in this case δDF = −LF . Proof: Suppose ﬁrst that F ∈ D1,2 and that DF belongs to Dom δ. Let G be a random variable in the nth chaos Hn . Then, applying Proposition 1.2.2 we have Jn F ). E(GδDF ) = E(DG, DF H ) = n2 (n − 1)! g, fn H ⊗n = nE(GJ Jn F , which implies F ∈ Dom L and δDF = −LF . So, Jn δDF = nJ Conversely, if F ∈ Dom L, then F ∈ D1,2 and for any G ∈ D1,2 , G =

∞ n=0 In (gn ), we have E(DG, DF H ) =

∞

nE(J Jn GJ Jn F ) = −E(GLF ).

n=1

Therefore, DF ∈ Dom δ, and δDF = −LF .

We are going to show that the operator L behaves as a second-order diﬀerential operator when it acts on smooth random variables.

60

1. Analysis on the Wiener space

Proposition 1.4.4 It holds that S ⊂ Dom L, and for any F ∈ S of the form F = f (W (h1 ), . . . , W (hn )), f ∈ Cp∞ (Rn ), we have LF

=

n

∂i ∂j f (W (h1 ), . . . , W (hn ))hi , hj H i,j=1 n −

∂i f (W (h1 ), . . . , W (hn ))W (hi ).

(1.70)

i=1

Proof:

We know that F belongs to D1,2 and that DF =

n

∂i f (W (h1 ), . . . , W (hn ))hi .

i=1

Consequently, DF ∈ SH ⊂ Dom δ and by Eq. (1.44) we obtain δDF

=

n

∂i f (W (h1 ), . . . , W (hn ))W (hi )

i=1 n

−

∂i ∂j f (W (h1 ), . . . , W (hn ))hi , hj H .

i,j=1

Now the result follows from Proposition 1.4.3. More generally, we can prove the following result.

Proposition 1.4.5 Suppose that F = (F 1 , . . . , F m ) is a random vector whose components belong to D2,4 . Let ϕ be a function in C 2 (Rm ) with bounded ﬁrst and second partial derivatives. Then ϕ(F ) ∈ Dom L, and L(ϕ(F )) =

m

∂i ∂j ϕ(F )DF i , DF j H +

i,j=1

m

∂i ϕ(F )LF i .

i=1

Proof: Approximate F by smooth random variables in the norm · 2,4 , and ϕ by functions in Cp∞ (Rm ), and use the continuity of the operator L in the norm · 2,2 . We can deﬁne on S the norm ! "1 F L = E(F 2 ) + E(|LF |2 ) 2 . Notice that Dom L = D2,2 and that the norms · L and · 2,2 coincide. In fact,

E(F 2 ) + E(|LF |2 )

=

∞

(n2 + 1)J Jn F 22

n=0

= E(F 2 ) + E(DF 2H ) + E(D2 F 2H⊗H ).

1.4 The Ornstein-Uhlenbeck semigroup

61

1,2 2 Similarly, the space √ D can be characterized as the domain in L (Ω) of the operator C = − −L deﬁned by

CF =

∞ √ − nJ Jn F. n=0

As in the case of the operator L, we can show that C is the inﬁnitesimal generator of a semigroup of operators (the Cauchy semigroup) given by Qt F =

∞

e−

√

nt

Jn F.

n=0

Observe that Dom C = D1,2 , and for any F ∈ Dom C we have E((CF )2 ) =

∞

nJ Jn F 22 = E(DF 2H ).

n=1

1.4.3 Hypercontractivity property and the multiplier theorem We have seen that Tt is a contraction operator on Lp (Ω) for any p ≥ 1. Actually, these operators verify a hypercontractivity property, which is due to Nelson [235]. In the next theorem this property will be proved using Itˆ oˆ’s formula, according to Neveu’s approach (cf. [236]). Theorem 1.4.1 Let p > 1 and t > 0, and set q(t) = e2t (p − 1) + 1 > p. Suppose that F ∈ Lp (Ω). Then T Tt F q(t) ≤ F p . Proof: Put q = q(t), and let q be the conjugate of q. Taking into account the duality between Lq (Ω) and Lq (Ω), it suﬃces to show that |E((T Tt F )G)| ≤ F p Gq for any F ∈ Lp (Ω) and for any G ∈ Lq (Ω). Tt F | ≤ Tt (|F |)), we may With the operator Tt nonnegative (which implies |T assume that F and G are nonnegative. By an approximation argument it suﬃces to suppose that there exist real numbers a ≤ b such that 0 < a ≤ F, G ≤ b < ∞. Also we may restrict our study to the case where F = f (W (h1 ), . . . , W (hn )) and G = g(W (h1 ), . . . , W (hn )) for some measurable functions f, g such that 0 < a ≤ f, g ≤ b < ∞ and orthonormal elements h1 , . . . , hn ∈ H. Let {β t , 0 ≤ t ≤ 1} and {ξ t , 0 ≤ t ≤ 1} be two independent Brownian motions. Consider orthonormal functions φ1 , . . . , φn ∈ L2 ([0, 1]). By (1.67) we can write 1 1 % φ1 dβ + 1 − e−2t φ1 dξ, E((T Tt F )G) = E f e−t . . . , e−t

1

φn dβ + 0

%

0

1 − e−2t 0

1

φn dξ g 0

0

1

φ1 dβ, . . . ,

1

φn dβ 0

.

62

1. Analysis on the Wiener space

In this way we can reduce our problem to show the following inequality: E(XY ) ≤ Xp Y q , where 0 < a ≤ X, Y ≤ b < ∞, and X, Y are random variables meato the σ-ﬁelds generated by the Brownian motions surable with respect √ η s = e−t β s + 1 − e−2t ξ s and β s , respectively. These random variables will have integral representations of the following kind: 1 1 p p q q ϕs dη s , Y = E(Y ) + ψ s dβ s . X = E(X ) + 0

0

Appling Itˆ o’s formula to the bounded positive martingales s s p q Ms = E(X ) + ϕu dη u and Ns = E(Y ) + ψ u dβ u , 0

0

and to the function f (x, y) = xα y γ , α = p1 , γ = XY

1 q ,

we obtain

= Xp Y q 1 α−1 γ α γ−1 + (αM Ms Ns dM Ms + γM Ms Ns dN Ns ) + 0

0

1

1 α γ M N As ds, 2 s s

where As = α(α − 1)M Ms−2 ϕ2s + γ(γ − 1)N Ns−2 ψ 2s + 2αγM Ms−1 Ns−1 ϕs ψ s e−t . Taking expectations, we get E(XY ) = Xp Y q +

1 2

1

E(M Msα Nsγ As )ds.

0

Therefore, it suﬃces to show that As ≤ 0. Note that α(α − 1) = p1 ( p1 − 1) < 0. Thus, As will be negative if α(α − 1)γ(γ − 1) − (αγe−t )2 ≥ 0. Finally, (α − 1)(γ − 1) − γαe−2t =

1 (p − 1 − (q − 1)e−2t ) = 0, pq

which achieves the proof. As a consequence of the hypercontractivity property it can be shown that for any 1 0 such that q = 1 + e2t (p − 1). Then for every F ∈ Hn we have e−nt F q = T Tt F q ≤ F p .

1.4 The Ornstein-Uhlenbeck semigroup

63

In addition, for each n ≥ 1 the operator Jn is bounded in Lp (Ω) for any 1 2 n J Jn F p ≤ (p − 1)− 2 F p if p < 2. In fact, suppose ﬁrst that p > 2, and let t > 0 be such that p−1 = e2t . Using the hypercontractivity property with the exponents p and 2, we obtain J Jn F p = ent T Tt Jn F p ≤ ent J Jn F 2 ≤ ent F 2 ≤ ent F p .

(1.71)

If p < 2, we use a duality argument: J Jn F p

=

sup E((J Jn F )G)

G q ≤1

≤

Jn Gq ≤ ent F p , F p sup J

G q ≤1

where q is the conjugate of p, and q − 1 = e2t . We are going to use the hypercontractivity property to show a multiplier theorem (see Meyer [225] and Watanabe [343]) that will be useful in proving Meyer’s inequalities. Recall that we denote by P the class of polynomial random variables. That means that a random variable F belongs to P if it is of the form F = p(W (h1 ), . . . , W (hn )), where h1 , . . . , hn are elements of H and p is a polynomial of n variables. The set P is dense in Lp (Ω) for all p ≥ 1 (see Exercise 1.1.7). Consider a sequence of real numbers {φ(n), n ≥ 0} with φ(0) = 0. This sequence determines a linear operator Tφ : P → P deﬁned by Tφ F =

∞

φ(n)J Jn F,

F ∈ P.

n=0

, L, and C √ are of this type, the corWe remark that the operators Tt , Qt√ responding sequences being e−nt , e− nt , −n, − n, respectively. We are interested in the following question: For which sequences is the operator Tφ bounded in Lp (Ω) for p > 1? Theorem 1.4.2 will give an answer to this problem. The proof of the multiplier theorem is based on the following technical lemma. Lemma 1.4.1 Let p > 1 and F ∈ P. Then for any integer N ≥ 1 there exists a constant K (depending on p and N ) such that T Tt (I − J0 − J1 − · · · − JN −1 )(F )p ≤ Ke−N t F p for all t > 0.

64

1. Analysis on the Wiener space

Proof: Assume ﬁrst that p > 2. Choose t0 such that p = e2t0 + 1. Then, by Nelson’s hypercontractivity theorem (Theorem 1.4.1) we have, for all t ≥ t0 , T Tt0 Tt−t0 (I − J0 − J1 − · · · − JN −1 )(F )2p ≤ T Tt−t0 (I − J0 − J1 − · · · − JN −1 )(F )22 ∞ ∞ = e−n(t−t0 ) Jn F 22 = e−2n(t−t0 ) J Jn F 22 n=N −2N (t−t0 )

≤e

F 22

n=N −2N (t−t0 )

≤e

F 2p ,

and this proves the desired inequality with K = eN t0 . For t < t0 , the inequality can be proved by the following direct argument, using (1.71): T Tt (I − J0 − J1 − · · · − JN −1 )(F )p ≤

N −1

J Jn F p + F p ≤

n=0

≤

2N t0

Ne

N t0

+e

N −1

ent0 F p + F p

n=0 −N t

e

F p .

For p = 2 the inequality is immediate, and for 1 < p < 2 it can be obtained by duality (see Exercise 1.4.5). The following is the multiplier theorem. a sequence of real numbers {φ(n), n ≥ 0} such Theorem 1.4.2 Consider

∞ that φ(0)

= 0 and φ(n) = k=0 ak n−k for n ≥ N and for some ak ∈ R ∞ −k < ∞. Then the operator such that k=0 |ak |N Tφ (F ) =

∞

φ(n)J Jn F

n=0

is bounded in Lp (Ω) for any 1 < p < ∞. Notice that the assumptions of this theorem are equivalent to saying that there exists a function h(x) analytic near the origin such that φ(n) = h(n−1 ) for n ≥ N . Proof:

Deﬁne

Tφ =

N −1 n=0

φ(n)J Jn +

∞ n=N

(1)

(2)

φ(n)J Jn = Tφ + Tφ .

1.4 The Ornstein-Uhlenbeck semigroup

65

(1)

We know that Tφ is bounded in Lp (Ω) because the operators Jn are bounded in Lp (Ω) for each ﬁxed n. We have

∞ ∞

(2)

−k ak n Jn F

Tφ F =

p n=N k=0 p

∞ ∞

≤ |ak | n−k Jn F . (1.72)

n=N

k=0

Now, using the equality k ∞ −k −nt n = e dt = [0,∞)k

0

p

e−n(t1 +···+tk ) dt1 · · · dtk

we obtain ∞

n−k Jn F =

[0,∞)k

n=N

Tt1 +···+tk (I − J0 − · · · − JN −1 )(F )dt1 · · · dtk .

Applying Lemma 1.4.1 yields

∞

−k n Jn F

n=N p ≤ T Tt1 +···+tk (I − J0 − · · · − JN −1 )(F )p dt1 · · · dtk [0,∞)k ≤ KF p e−N (t1 +···+tk ) dt1 · · · dtk [0,∞)k

= KN

−k

F p ,

(1.73)

where the constant K depends only on p and N . Substituting (1.73) into (1.72) we obtain ∞

(2) |ak |N −k F p ,

Tφ F ≤ K p

k=0

which allows us to complete the proof.

For example, the operator Tφ = (I − L)−α deﬁned by the sequence α > 0, is bounded in Lp (Ω), for 1 < p < ∞, φ(n) = (1 + n)−α , where because h(x) =

x x+1

α

is analytic in a neibourhood of the origin. Actually

this operator is a contraction in Lp (Ω) for any 1 ≤ p < ∞ (see Exercise 1.4.8). The following commutativity relationship holds for a multiplier operators Tφ .

66

1. Analysis on the Wiener space

Lemma 1.4.2 Consider a sequence of real numbers {φ(n), n ≥ and the

0} ∞ associated linear operator Tφ from P into P. Deﬁne Tφ+ = n=0 φ(n + 1)J Jn . Then for any F ∈ P it holds that DT Tφ (F ) = Tφ+ D(F ).

(1.74)

Proof: Without loss of generality we can assume that F belongs to the nth Wiener chaos Hn , n ≥ 0. In that case we have DT Tφ (F ) = D(φ(n)F ) = φ(n)DF = Tφ+ D(F ).

Exercises 1.4.1 Let W = {W Wt , t ≥ 0} be a standard Brownian motion. Check that the process % t ∈ R, Yt = βe−αt W (e2αt ), has the law of an Ornstein-Uhlenbeck process with parameters α, β. 1.4.2 Suppose that (Ω, F, P ) is the classical Wiener space (that is, Ω = Tt , t ≥ 0} be the OrnsteinC0 ([0, 1]) and P is the Wiener measure). Let {T Uhlenbeck semigroup given by % (T Tt F )(u) = F (e−t u + 1 − e−2t ω)P (dω), Ω

for all F ∈ L2 (Ω). Consider a Brownian measure W on [0, 1] × R+ , deﬁned F, P), and with Lebesgue measure as control on some probability space (Ω, measure. Then W (s, t) = W ([0, s] × [0, t]), (s, t) ∈ [0, 1] × R+ , is a twoparameter Wiener process that possesses a continuous version. Deﬁne X(t, τ ) = e−t W (τ , e2t ),

t ∈ R,

τ ∈ [0, 1].

Compute the covariance function of X. Show that Xt = X(t, ·) is a Ω-valued F, P) stationary continuous Markov process on the probability space (Ω, such that it admits Tt as semigroup of operators. Hint: Use the arguments of Proposition 1.4.1’s proof to show that F (Xs+t )|Fe2s = (T Tt F )(Xs ) E for all t ≥ 0, s ∈ R, F ∈ L2 (Ω), where F˜t , t ≥ 0, is the σ-ﬁeld generated by the random variables {W (τ , σ), 0 ≤ σ ≤ t, τ ∈ [0, 1]}.

∞ F1− − 1.4.3 For any 0 < < 1 put F1− = n=0 (1 − )n Jn F and F = 1 [F F ]. Show that LF exists if and only if F converges in L2 (Ω) as ↓ 0, and in this case LF = lim↓0 F .

1.5 Sobolev spaces and the equivalence of norms

67

1.4.4 Set F = exp(W (h) − 12 h2H ), h ∈ H. Show that LF = −(W (h) − h2H )F . 1.4.5 Complete the proof of Lemma 1.4.1 in the case 1 < p < 2, using a duality argument. 1.4.6 Using the Gaussian formula (A.1), show that the multiplier theorem (Theorem 1.4.2) is still valid for Hilbert-valued random variables. 1.4.7 Show that the operator L is local in the domain Dom L. That is, LF 1{F =0} = 0 for any random variable F in Dom L. 1.4.8 Show that the operator (I − L)−α is a contraction in Lp (Ω) for any 1 ≤ p < ∞, where α > 0. ∞ Hint: Use the equation (1 + n)−α = Γ(α)−1 0 e−(n+1)α tα−1 dt. 1.4.9 Show that if F ∈ D1,2 and G is a square integrable random variable such that E(G) = 0, then E(F G) = E

#

$ DF, DC −2 G H .

1.5 Sobolev spaces and the equivalence of norms In this section we establish Meyer’s inequalities, following the method of Pisier [285]. Let V be a Hilbert space. We recall that the spaces Dk,p (V ), for any integer k ≥ 1 and any real number p ≥ 1 have been deﬁned as the completion of the family of V -valued smooth random variables SV with respect to the norm · k,p,V deﬁned in (1.37). Consider the intersection D∞ (V ) = ∩p≥1 ∩k≥1 Dk,p (V ). Then D∞ (V ) is a complete, countably normed, metric space. We will write D∞ (R) = D∞ . For every integer k ≥ 1 and any real number p ≥ 1 the operator D is continuous from Dk,p (V ) into Dk−1,p (H ⊗ V ). Consequently, D is a continuous linear operator from D∞ (V ) into D∞ (H ⊗ V ). Moreover, if F and G are random variables in D∞ , then the scalar product DF, DGH is also in D∞ . The following result can be easily proved by approximating the components of the random vector F by smooth random variables. Proposition 1.5.1 Suppose that F = (F 1 , . . . , F m ) is a random vector whose components belong to D∞ . Let ϕ ∈ Cp∞ (Rm ). Then ϕ(F ) ∈ D∞ , and we have m ∂i ϕ(F )DF i . D(ϕ(F )) = i=1

68

1. Analysis on the Wiener space

In particular, we deduce that D∞ is an algebra. We will see later that L is a continuous operator from D∞ into D∞ and that the operator δ is continuous from D∞ (H) into D∞ . To show these results we will need Meyer’s inequalities, which provide the equivalence between the p norm of CF and that √ of DF H for p > 1 (we recall that C is the operator deﬁned by C = − −L). This equivalence of norms will follow from the fact that the operator DC −1 is bounded in Lp (Ω) for any p > 1, and this property will be proved using the approach by Pisier [285] based on the boundedness in Lp of the Hilbert transform. We recall that the Hilbert transform of a function f ∈ C0∞ (R) is deﬁned by f (x + t) − f (x − t) dt. Hf (x) = t R The transformation H is bounded in Lp (R) for any p > 1 (see Dunford and Schwarz [87], Theorem XI.7.8). Consider the function ϕ : [− π2 , 0) ∪ (0, π2 ] → R+ deﬁned by 1 1 ϕ(θ) = √ |π log cos2 θ|− 2 sign θ. 2

(1.75)

1 Notice that when θ is close to zero this function tends to inﬁnity as √2πθ . Suppose that {W (h), h ∈ H} is an independent copy of the Gaussian process {W (h), h ∈ H}. We will assume as in Section 1.4 that W and W are deﬁned in the product probability space (Ω × Ω , F ⊗ F , P × P ). For Wθ (h), h ∈ H} deﬁned by any θ ∈ R we consider the process Wθ = {W

Wθ (h) = W (h) cos θ + W (h) sin θ,

h ∈ H.

This process is Gaussian, with zero mean and with the same covariance function as {W (h), h ∈ H}. Let W : Ω → RH and W : Ω → RH be the canonical mappings associated with the processes {W (h), h ∈ H} and {W (h), h ∈ H}, respectively. Given a random variable F ∈ L0 (Ω, F, P ), we can write F = ψ F ◦ W , where ψ F is a measurable mapping from RH to R, determined P ◦ W −1 a.s. As a consequence, the random variable Wθ ) = ψ F (W cos θ + W sin θ) is well deﬁned P × P a.s. We set ψ F (W Rθ F = ψ F (W Wθ ).

(1.76)

We denote by E the mathematical expectation with respect to the probability P , and by D the derivative operator with respect to the Gaussian process W (h). With these notations we can write the following expression for the operator D(−C)−1 . Lemma 1.5.1 For every F ∈ P such that E(F ) = 0 we have π2 D(−C)−1 F = E (D (Rθ F ))ϕ(θ)dθ. −π 2

(1.77)

1.5 Sobolev spaces and the equivalence of norms

69

Proof: Suppose that F = p(W (h1 ), . . . , W (hn )), where h1 , . . . , hn ∈ H and p is a polynomial in n variables. We have Rθ F = p(W (h1 ) cos θ + W (h1 ) sin θ, . . . , W (hn ) cos θ + W (hn ) sin θ), and therefore

D (Rθ F ) =

n

∂i p(W (h1 ) cos θ + W (h1 ) sin θ,

i=1

. . . , W (hn ) cos θ + W (hn ) sin θ) sin θhi (s) = sin θRθ (DF ). Consequently, using Mehler’s formula (1.67) we obtain E (D (Rθ F )) = sin θE (Rθ (DF )) = sin θT Tt (DF ), where t > 0 is such that cos θ = e−t . This implies E (D (Rθ F )) =

∞

sin θ(cos θ)n Jn DF.

n=0

Note that since F is a polynomial random variable the above series is actually the sum of a ﬁnite number of terms. By Exercise 1.5.3, the righthand side of (1.77) can be written as ∞ n=0

π 2

−π 2

n

sin θ(cos θ) ϕ(θ)dθ Jn DF =

∞

√

n=0

1 Jn DF. n+1

Finally, applying the commutativity relationship (1.74) to the multiplication operator deﬁned by the sequence φ(n) = √1n , n ≥ 1, φ(0) = 0, we get Tφ F = D(−C)−1 F, Tφ+ DF = DT and the proof of the lemma is complete.

Now with the help of the preceding equation we can show that the operator DC −1 is bounded from Lp (Ω) into Lp (Ω; H) for any p > 1. Henceforth cp and Cp denote generic constants depending only on p, which can be diﬀerent from one formula to another. Proposition 1.5.2 Let p > 1. There exists a ﬁnite constant cp > 0 such that for any F ∈ P with E(F ) = 0 we have DC −1 F p ≤ cp F p .

70

Proof:

1. Analysis on the Wiener space

Using (1.77) we can write

p E DC −1 F H

π

p

2

=E E (D (Rθ F ))ϕ(θ)dθ

−π

2 H p π 2 −1 = αp EE W E (D (Rθ F ))ϕ(θ)dθ , π − 2

where αp = E(|ξ|p ) with ξ an N (0, 1) random variable. We recall that by Exercise 1.2.6 (Stroock’s formula) for any G ∈ L2 (Ω , F , P ) the Gaussian random variable W (E (D G)) is equal to the projection J1 G of G on the ﬁrst Wiener chaos. Therefore, we obtain that

p E DC −1 F H p π 2 J1 Rθ F ϕ(θ)dθ = α−1 p EE −π 2 p π2 = α−1 Rθ F ϕ(θ)dθ J1 p.v. p EE π −2 π2 p ≤ cp EE p.v. Rθ F ϕ(θ)dθ , −π 2

for some constant cp > 0 (where the abbreviation p.v. stands for principal value). Notice that the function Rθ F ϕ(θ) might not belong to L1 (− π2 , π2 ) because, unlike the term J1 Rθ F , the function Rθ F may not balance the singularity of ϕ(θ) at the origin. For this reason we have to introduce the principal value integral p.v.

π 2

−π 2

Rθ F ϕ(θ)dθ = lim ↓0

≤|θ|≤ π 2

Rθ F ϕ(θ)dθ,

which can be expressed as a convergent integral in the following way: 0

π 2

[Rθ F ϕ(θ) + R−θ F ϕ(−θ)]dθ = 0

π 2

[R F − R−θ F ] %θ dθ. 2π| log cos2 θ|

1.5 Sobolev spaces and the equivalence of norms

71

For any ξ ∈ R we deﬁne the process Rξ (h) = (W (h) cos ξ + W (h) sin ξ, −W (h) sin ξ + W (h) cos ξ). The law of this process is the same as that of {(W (h), W (h)), h ∈ H}. On the other hand, Rξ Rθ F = Rξ+θ F , where we set Rξ G((W (h1 ), W (h1 )), . . . , (W (hn ), W (hn ))) = G(Rξ (h1 ), . . . , Rξ (hn )). Therefore, we get

π2

Rθ F ϕ(θ)dθ

p.v. π

− 2

p

( ) π2

= Rξ p.v. Rθ F ϕ(θ)dθ π

−2 p

π2

= p.v. Rξ+θ F ϕ(θ)dθ , π

− 2

p

where · p denotes the L norm with respect to P × P . Integration with respect to ξ yields p π π2 2

p −1 Rξ+θ F ϕ(θ)dθ dξ . (1.78) E DC F H ≤ cp EE p.v. π π − − p

2

2

Furthermore, there exists a bounded continuous function ϕ and a constant c > 0 such that c ϕ(θ) = ϕ (θ) + , θ on [− π2 , π2 ]. Consequently, using the Lp boundedness of the Hilbert transform, we see that the right-hand side of (1.78) is dominated up to a constant by π 2 p EE |Rθ F | dθ = πF pp . −π 2

In fact, the term ϕ (θ) is easy to treat. On the other hand, to handle the term θ1 it suﬃces to write p π2 π2 Rξ+θ F − Rξ−θ F dθ dξ −π θ −π 2 2 p R ξ+θ F − R ξ−θ F dθ dξ ≤ cp θ R R π2 Rξ+θ F p + θ dθdξ π −π [−2π,− π 2 2 ]∪[ 2 ,2π] π2 2π p p ≤ cp |Rθ F | dθ + |Rξ+θ F | dθdξ , R

θ F = 1[− 3π , 3π ] (θ)Rθ F . where R 2 2

−π 2

−2π

72

1. Analysis on the Wiener space

Proposition 1.5.3 Let p > 1. Then there exist positive and ﬁnite constants cp and Cp such that for any F ∈ P we have cp DF Lp (Ω;H) ≤ CF p ≤ Cp DF Lp (Ω;H) .

(1.79)

Proof: We can assume that the random variable F has zero expectation. Set G = CF . Then, using Proposition 1.5.2, we have DF Lp (Ω;H) = DC −1 GLp (Ω;H) ≤ cp Gp = cp CF p , which shows the left inequality. We will prove the right inequality using a = C −1 (I − J0 )(G), and denote the duality argument. Let F, G ∈ P. Set G conjugate of p by q. Then we have |E(GCF )|

= |E(DF, DG H )| = |E((I − J0 )(G)CF )| = |E(CF C G)| Lq (Ω;H) ≤ cq DF Lp (Ω;H) C G q ≤ DF Lp (Ω;H) DG

= cq DF Lp (Ω;H) (I − J0 )(G)q ≤ cq DF Lp (Ω;H) Gq . Taking the supremum with respect to G ∈ P with Gq ≤ 1, we obtain CF p ≤ cq DF Lp (Ω;H) .

Now we can state Meyer’s inequalities in the general case. Theorem 1.5.1 For any p > 1 and any integer k ≥ 1 there exist positive and ﬁnite constants cp,k and Cp,k such that for any F ∈ P, cp,k E Dk F pH ⊗k

≤ E |C k F |p ! " ≤ Cp,k E Dk F pH ⊗k + E(|F |p ) .

(1.80)

Proof: The proof will be done by induction on k. The case k = 1 is included in Proposition 1.5.3. Suppose that the left-hand side of (1.80) holds for 1, . . . , k. Consider two families of independent random variables, with the identical distribution N (0, 1), deﬁned in the probability space ([0, 1], B([0, 1]), λ) (λ is the Lebesgue measure) {γ α (s), s ∈ [0, 1], α ∈ Nk∗ }, where N∗ = {1, 2, . . . } and {γ i (s), s ∈ [0, 1], i ≥ 1}. Suppose that F = p(W (h1 ), . . . , W (hn )), where the hi ’s are orthonormal elements of H. We ﬁx a complete orthonormal system {ei , i ≥ 1} in H which contains the hi ’s. We set Di (F ) = DF, ei H and Dαk (F ) = Dα1 Dα2 · · · Dαk (F ) for any multiindex α = (α1 , . . . , αk ). With these notations, using the Gaussian

1.5 Sobolev spaces and the equivalence of norms

73

formula (A.1) and Proposition 1.5.3, we can write

p E Dk+1 F H ⊗(k+1) ⎛ p2 ⎞ ∞ 2 ⎟ ⎜ k Di Dα F ⎠ = E ⎝ i=1 α∈Nk ∗ p ⎞ ⎛ 1 1 ∞ k ⎠ dtds ⎝ = A−1 E D D F γ (t)γ (s) i α α i p 0 0 i=1 α∈Nk ∗ ⎛ p⎞ ⎡ ⎛ ⎞⎤2 2 1 ∞ ⎟ ⎜ ⎟ k ⎣Di ⎝ ⎠ ⎦ ≤ E⎜ D F γ (t) ⎠ dt α α ⎝ 0 i=1 α∈Nk ∗ ⎛ ⎛ ⎞ p ⎞ 1 ≤ cp E ⎝C ⎝ Dαk F γ α (t)⎠ ⎠ dt 0 α∈NN ∗ ⎛ p2 ⎞ 2 ⎟ ⎜ ≤ cp E ⎝ CDαk F ⎠ . α∈Nk ∗

Consider the operator Rk (F ) =

∞ n=k

, 1−

k Jn F, n

F ∈ P.

By Theorem 1.4.2 this operator is bounded in Lp (Ω), and using the induction hypothesis we can write ⎛ ⎛ p2 ⎞ p2 ⎞ 2 ⎟ 2 ⎟ ⎜ ⎜ CDαk F ⎠ = E ⎝ Dαk CRk F ⎠ E ⎝ α∈Nk α∈Nk ∗ ∗

p = E Dk CRk F H ⊗k p ≤ cp,k E C k+1 Rk F p ≤ cp,k E C k+1 F for some constant cp,k > 0. We will prove by induction the right inequality in (1.80) for F ∈ P satisfying (J J0 + J1 + · · · + Jk−1 )(F ) = 0. The general case would follow easily (Exercise 1.5.1). Suppose that this holds for k. Applying Proposition

74

1. Analysis on the Wiener space

1.5.3 and the Gaussian formula (A.1), we have

p E C k+1 F

⎛ p2 ⎞ ∞ 2 Di C k F ⎠ ≤ cp E DC k F pH = cp E ⎝ i=1 1 ∞ p −1 k = Ap cp Di C F γ i (s) ds. E 0 i=1

Consider the operator deﬁned by

Rk,1

k ∞ , n = Jn . n−1 n=2

Using the commutativity relationship (1.74), our induction hypothesis, and the Gaussian formula (A.1), we can write p ∞ k Di C F γ i (s) ds E 0 i=1 p 1 ∞ k = C Di Rk,1 F γ i (s) ds E 0 i=1

p 1 ∞

k

≤ Cp,k E D (Di Rk,1 F ) γ i (s) ds

⊗k 0 i=1 H ⎞ ⎛ ∞ 2 p2 1 k ⎟ ⎜ = Cp,k Dα Di Rk,1 F γ i (s) ⎠ ds E ⎝ 0 α∈Nk i=1 ∗ p ⎞ ⎛ 1 1 ∞ k ⎠ dsdt. ⎝ D E D R F γ (s)γ (t) = Cp,k A−1 i α p α i N,1 0 0 α∈Nk i=1

1

∗

Finally, if we introduce the operator ∞ n+1+k 2

k

Rk,2 =

n=0

n+k

Jn ,

1.5 Sobolev spaces and the equivalence of norms

75

we obtain, by applying the commutativity relationship, the Gaussian formula (A.1), and the boundedness in Lp (Ω) of the operator Rk,2 , that p ⎞ ⎛ 1 1 ∞ k Dα Di Rk,1 F γ i (s)γ α (t) ⎠ dsdt E ⎝ 0 0 α∈Nk i=1 ∗ p ⎞ ⎛ 1 1 ∞ = Rk,2 Dαk Di F γ i (s)γ α (t) ⎠ dsdt E ⎝ 0 0 α∈Nk i=1 ∗ p ⎞ ⎛ 1 1 ∞ k ≤ Cp,k Dα Di F γ i (s)γ α (t) ⎠ dsdt E ⎝ 0 0 α∈Nk i=1 ∗ ⎛ p2 ⎞ ∞ 2 ⎟ ⎜ k = Cp,k Ap E ⎝ Dα Di F ⎠ α∈Nk i=1 ∗ = Cp,k Ap E Dk+1 F pH ⊗(k+1) , which completes the proof of the theorem. The inequalities (1.80) also hold for polynomial random variables taking values in a separable Hilbert space (see Execise 1.5.5). One of the main applications of Meyer’s inequalities is the following result on the continuity of the operator δ. Here we consider δ as the adjoint of the derivative operator D on Lp (Ω). Proposition 1.5.4 The operator δ is continuous from D1,p (H) into Lp (Ω) for all p > 1. Proof: Let q be the conjugate of p. For any u in D1,p (H) and any polynomial random variable G with E(G) = 0 we have E(δ(u)G) = E(u, DGH ) = E(u, ˜ DGH ) + E(E(u), DGH ), where u ˜ = u − E(u). Notice that the second summand in the above expression can be bounded by a constant times uLp (Ω;H) Gq . So we can assume E(u) = E(DG) = 0. Then we have, using Exercise 1.4.9 |E(δ(u)G)|

= |E(u, DGH )| = |E(Du, DC −2 DGH⊗H )| ≤

DuLp (Ω;H⊗H) DC −2 DGLq (Ω;H⊗H)

≤ cp DuLp (Ω;H⊗H) D2 C −2 RGLq (Ω;H⊗H) ≤ cp DuLp (Ω;H⊗H) Gq , where R=

∞

n Jn , n − 1 n=2

76

1. Analysis on the Wiener space

and we have used Meyer’s inequality and the boundedness in Lq (Ω) of the operator R. So, we have proved that δ is continuous from D1,p (H) into Lp (Ω). Consider the set PH of H-valued polynomial random variables. We have the following result: Lemma 1.5.2 For any process u ∈ PH and for any p > 1, we have C −1 δ(u)p ≤ cp uLp (Ω;H) . Proof: Let G ∈ P with E(G) = 0 and u ∈ PH . Using Proposition 1.5.3 we can write |E(C −1 δ(u) G)|

= |E(u, DC −1 GH )| ≤ uLp (Ω;H) DC −1 GLq (Ω;H) ≤ cp uLp (Ω;H) GLq (Ω) ,

where q is the conjugate of p. This yields the desired estimation. −1 p As a consequence, the operator D(−L) δ is bounded from L (Ω; H) into Lp (Ω; H). In fact, we can write D(−L)−1 δ = [DC −1 ][C −1 δ]. Using Lemma 1.5.2 we can show the following result: Proposition 1.5.5 Let F be a random variable in Dk,α with α > 1. Suppose that Di F belongs to Lp (Ω; H ⊗i ) for i = 0, 1, . . . , k and for some p > α. Then F ∈ Dk,p , and there exists a sequence Gn ∈ P that converges to F in the norm · k,p . Proof: We will prove the result only for k = 1; a similar argument can be used for k > 1. We may assume that E(F ) = 0. We know that PH is dense in Lp (Ω; H). Hence, we can ﬁnd a sequence of H-valued polynomial random variables η n that converges to DF in Lp (Ω; H). Without loss of generality J0 ) we may assume that Jk η n ∈ PH for all k ≥ 1. Note that −L−1 δD = (I −J on D1,α . Consider the decomposition η n = DGn + un given by Proposition 1.3.10. Notice that Gn ∈ P because Gn = −L−1 δ(η n ) and δ(un ) = 0. Using the boundedness in Lp of the operator C −1 δ (which implies that of L−1 δ by Exercise 1.4.8), we obtain that F − Gn = L−1 δ(η n − DF ) converges to zero in Lp (Ω) as n tends to inﬁnity. On the other hand, DF − DGn Lp (Ω;H) = DL−1 δ(η n − DF )Lp (Ω;H) ≤ cp η n − DF Lp (Ω;H) ; hence, DGn − DF H converges to zero in Lp (Ω) as n tends to inﬁnity. So the proof of the proposition is complete. Corollary 1.5.1 The class P is dense in Dk,p for all p > 1 and k ≥ 1.

1.5 Sobolev spaces and the equivalence of norms

77

As a consequence of the above corollary, Theorem 1.5.1 holds for random k variables in Dk,p , and the operator (−C)k = (−L) 2 is continuous from Dk,p into Lp . Thus, L is a continuous operator on D∞ . The following proposition is a H¨ o¨lder inequality for the ·k,p norms. Proposition 1.5.6 Let F ∈ Dk,p , G ∈ Dk,q for k ∈ N∗ , 1 < p, q < ∞ and let r be such that p1 + 1q = 1r . Then, F G ∈ Dk,r and F Gk,r ≤ cp,q,k F k,p Gk,q . Proof: Suppose that F, G ∈ P. By Leipnitz rule (see Exercise 1.2.13) we can write Dk (F G) =

k

k

Di F ⊗i Dk−i G ⊗(k−i) . H H i i=0

Hence, by H¨¨older’s inequality F Gk,r

≤

j k

j

Di F ⊗i

Dj−i G ⊗(j−i) H H p q i j=0 i=0

≤ cp,q,k F k,p Gk,q . We will now introduce the continuous family of Sobolev spaces deﬁned by Watanabe (see [343]). For any p > 1 and s ∈ R we will denote by |·|s,p the seminorm

s |F |s,p = (I − L) 2 F p , s

where F is a polynomial random variable. Note that (I −L) 2 F = s n) 2 Jn F. These seminorms have the following properties:

∞

n=0 (1+

(i) |F |s,p is increasing in both coordinates s and p. The monotonicity s in p is clear and in s follows from the fact that the operators (I − L) 2 are contractions in Lp for all s < 0, p > 1 (see Exercise 1.4.8). (ii) The seminorms |·|s,p are compatible, in the sense that for any sequence Fn in P converging to zero in the norm |·|s,p , and being a Cauchy sequence in another norm |·|s ,p , it also converges to zero in the norm |·|s ,p . For any p > 1, s ∈ R, we deﬁne Ds,p as the completion of P with respect to the norm |·|s,p .

78

1. Analysis on the Wiener space

Remarks: 1. |F |0,p = F 0,p = F p , and D0,p = Lp (Ω). For k = 1, 2, . . . the seminorms |·|k,p and ·k,p are equivalent due to Meyer’s inequalities. In fact, we have

k k

|F |k,p = (I − L) 2 F p ≤ |E(F )| + R(−L) 2 F , p

k2

∞ where R = n=1 n+1 Jn . By Theorem 1.4.2 this operator is bounded n in Lp (Ω) for all p > 1. Hence, applying Theorem 1.5.1 we obtain

k

2 |F |k,p ≤ ck,p F p + (−L) F p

≤ ck,p F p + Dk F Lp (Ω;H ⊗k ) ≤ ck,p F k,p . In a similar way one can show the converse inequality (Exercise 1.5.9). Thus, by Corollary 1.5.1 the spaces Dk,p coincide with those deﬁned using the derivative operator.

2. From properties (i) and (ii) we have Ds,p ⊂ Ds ,p if p ≤ p and s ≤ s. 3. For s > 0 the operator (I − L)− 2 is an isometric isomorphism (in the norm |·|s,p ) between Lp (Ω) and Ds,p and between D−s,p and Lp (Ω) for all p > 1. As a consequence, the dual of Ds,p is D−s,q where p1 + 1q = 1. If s < 0 the elements of Ds,p may not be ordinary random variables and they are interpreted as distributions on the Gaussian space or generalized random variables. Set D−∞ = ∪s,p Ds,p . The space D−∞ is the dual of the space D∞ which is a countably normed space. The interest of the space D−∞ is that it contains the composition of Schwartz distributions with smooth and nondegenerate random variables, as we shall show in the next chapter. An example of a distribution random variable is the compostion δ 0 (W (h)) (see Exercise 1.5.6). s

4. Suppose that V is a real separable Hilbert space. We can deﬁne the Sobolev spaces Ds,p (V ) of V -valued functionals as the completion of the class PV of V -valued polynomial random variables with respect to the seminorm |·|s,p,V deﬁned in the same way as before. The above properties are still true for V -valued functionals. If F ∈ Ds,p (V ) and G ∈ D−s,q (V ), where p1 + 1q = 1, then we denote the pairing F, G by E(F, GV ). Proposition 1.5.7 Let V be a real separable Hilbert space. For every p > 1 and s ∈ R, the operator D is continuous from Ds,p (V ) to Ds−1,p (V ⊗H) and the operator δ (deﬁned as the adjoint of D) is continuous from Ds,p (V ⊗H) into Ds−1,p (V ). That is, for all p > 1 and s ∈ R, we have |δ(u)|s−1,p ≤ cs,p |u|s,p,H .

1.5 Sobolev spaces and the equivalence of norms

79

Proof: For simplicity we assume that V = R. Let us prove ﬁrst the continuity of D. For any F ∈ P we have s

s

(I − L) 2 DF = DR(I − L) 2 F, where

2 ∞ n Jn . n+1 n=1 s

R=

By Theorem 1.4.2 the operator R is bounded in Lp (Ω) for all p > 1, and we obtain

s s |DF |s+1,p,H = (I − L) 2 DF Lp (Ω;H) = DR(I − L) 2 F Lp (Ω;H)

s s ≤ R(I − L) 2 F 1,p ≤ cp R(I − L) 2 F 1,p

s+1 1 s

= cp (I − L) 2 R(I − L) 2 F = cp R(I − L) 2 F p p

s+1

≤ cp (I − L) 2 F = cp |F |s+1,p . p

The continuity of the operator δ follows by a duality argument. In fact, for any u ∈ Ds,p (H) we have |δ(u)|s−1,p

=

sup

|F | 1−s,q ≤1

|E (u, DF H )| ≤ |u|s,p,H |DF |−s,q,H

≤ cs,p |u|s,p,H . Proposition 1.5.7 allows us to generalize Lemma 1.2.3 in the following way: Lemma 1.5.3 Let {F Fn , n ≥ 1} be a sequence of random variables convergFn |s,p < ∞ for ing to F in Lp (Ω) for some p > 1. Suppose that supn |F some s > 0. Then Then F belongs to Ds,p . Proof:

We know that

s sup (I − L) 2 Fn p < ∞. n

Let q be the conjugate of p. There exists a subsequence {F Fn(i) , i ≥ 1} such s that (I − L) 2 Fn(i) converges weakly in σ(Lp , Lq ) to some element G. Then for any polynomial random variable Y we have s s E F (I − L) 2 Y = lim E Fn(i) (I − L) 2 Y n s = lim E (I − L) 2 Fn(i) Y = E(GY ). n

− 2s

Thus, F = (I − L) G, and this implies that F ∈ Ds,p . The following proposition provides a precise estimate for the norm p of the divergence operator.

80

1. Analysis on the Wiener space

Proposition 1.5.8 Let u be an element of D1,p (H), p > 1. Then we have δ(u)p ≤ cp E(u)H + DuLp (Ω;H⊗H) . Proof: From Proposition 1.5.7 we know that δ is continuous from D1,p (H) into Lp (Ω). This implies that δ(u)p ≤ cp uLp (Ω;H) + DuLp (Ω;H⊗H) . On the other hand, we have uLp (Ω;H) ≤ E(u)H + u − E(u)Lp (Ω;H) , and

1

= (I − L)− 2 RCu

u − E(u)Lp (Ω;H)

Lp (Ω;H)

≤ cp CuLp (Ω;H)

≤ cp DuLp (Ω;H⊗H) , where R =

∞

n=1 (1

1

+ n1 ) 2 Jn .

Exercises 1.5.1 Complete the proof of Meyer’s inequality (1.80) without the condition (J J0 + · · · + JN −1 )(F ) = 0. 1.5.2 Derive the right inequality in (1.80) from the left inequality by means of a duality argument. 1.5.3 Show that 0

π 2

sin θ cosn θ 1 % dθ = % . 2 π| log cos θ| 2(n + 1) 2

Hint: Change the variables, substituting cos θ = y and y = exp(− x2 ). 1.5.4 Let W = {W Wt , t ∈ [0, 1]} be a Brownian motion. For every 0 < γ < 1 and p = 2, 3, 4, . . . such that γ < 12 − 2p , we deﬁne the random variable W 2p p,γ =

[0,1]2

1 2

|W Ws − Wt |2p dsdt. |s − t|1+2pγ

∞ (see Airault and Malliavin [3]). Show that W 2p p,γ belongs to D

1.5.5 Using the Gaussian formula (A.1), extend Theorem 1.5.1 to a polynomial random variable with values on a separable Hilbert space V (see Sugita [323]).

1.5 Sobolev spaces and the equivalence of norms

81

1.5.6 Let p (x) be the density of the normal distribution N (0, ), for any > 0. Fix h ∈ H. Using Stroock’s formula (see Exercise 1.2.6) and the expression of the derivatives of p (x) in terms of Hermite polynomials, show the following chaos expansion: p (W (h)) =

∞ m=0

√

(−1)m I2m (h⊗2m ) m+ 12 . 2π 2m m ! h 2H +

Letting tend to zero in the above expression, ﬁnd the chaos expansion of δ 0 (W (h)) and deduce that δ 0 (W (h)) belongs to the negative Sobolev space 1 D−α,2 for any α > 12 , and also that δ 0 (W (h)) is not in D− 2 ,2 . 1.5.7 (See Sugita [325]) Let F be a smooth functional of a Gaussian process {W (h), h ∈ H}. Let {W (h), h ∈ H} be an independent copy of {W (h), h ∈ H}. a) Prove the formula D(T Tt F ) = √

% e−t E (D (F (e−t W + 1 − e−2t W ))) −2t 1−e

for all t > 0, where D denotes the derivative operator with respect to W . b) Using part a), prove the inequality p e−t p E(|F |p ), E(D(T Tt F )H ) ≤ cp √ 1 − e−2t for all p > 1. c) Applying part b), show that the operator (−L)k Tt is bounded in Lp and that Tt is continuous from Lp into Dk,p , for all k ≥ 1 and p > 1. 1.5.8 Prove Proposition 1.5.7 for k > 1. 1.5.9 Prove that |F |k,p ≤ ck,p F k,p for all p > 1, k ∈ N and F ∈ P.

Notes and comments [1.1] The notion of Gaussian space or the isonormal Gaussian process was introduced by Segal [303], and the orthogonal decomposition of the space of square integrable functionals of the Wiener process is due to Wiener [349]. We are interested in results on Gaussian families {W (h), h ∈ H} that depend only on the covariance function, that is, on the underlying Hilbert space H. One can always associate to the Hilbert space H an abstract Wiener space (see Gross [128]), that is, a Gaussian measure µ on a Banach space Ω such that H is injected continuously into Ω and 1 exp(ity, x)µ(dy) = x2H 2 Ω

82

1. Analysis on the Wiener space

for any x ∈ Ω∗ ⊂ H. In this case the probability space has a nice topological structure, but most of the notions introduced in this chapter are not related to this structure. For this reason we have chosen an arbitrary probability space as a general framework. For the deﬁnition and properties of multiple stochastic integrals with respect to a Gaussian measure we have followed the presentation provided by Itoˆ in [153]. The stochastic integral of adapted processes with respect to the Brownian motion originates in Itoˆ [152]. In Section 1.1.3 we described some elementary facts about the Itˆo integral. For a complete exposition of this subject we refer to the monographs by Ikeda and Watanabe [146], Karatzas and Shreve [164], and Revuz and Yor [292]. [1.2] The derivative operator and its representation on the chaotic development has been used in diﬀerent frameworks. In the general context of a Fock space the operator D coincides with the annihilation operator studied in quantum probability. The notation Dt F for the derivative of a functional of a Gaussian process has been taken from the work of Nualart and Zakai [263]. The bilinear form (F, G) → E(DF, DGH ) on the space D1,2 is a particular type of a Dirichlet form in the sense of Fukushima [113]. In this sense some of the properties of the operator D and its domain D1,2 can be proved in the general context of a Dirichlet form, under some additional hypotheses. This is true for the local property and for the stability under Lipschitz maps. We refer to Bouleau and Hirsch [46] and to Ma and R¨ o¨ckner [205] for monographs on this theory. In [324] Sugita provides a characterization of the space D1,2 in terms of diﬀerentiability properties. More precisely, in the case of the Brownian motion, a random variable F ∈ L2 (Ω) belongs to D1,2 if and only if the following two conditions are satisﬁed: (i) F is ray absolutely continuous (RAC). This means · that for any h ∈ H there exists a version of the process {F (ω + t 0 hs ds), t ∈ R} that is absolutely continuous. (ii) There exists a random vector DF ∈ L2 (Ω; H) such that for any · 1 h ∈ H, t [F (ω+t 0 hs ds)−F (ω)] converges in probability to DF, hH as t tends to zero. In Lemma 2.1.5 of Chapter 2 we will show that properties (i) and (ii) hold for any random variable F ∈ D1,p , p > 1. Proposition 1.2.6 is due to Sekiguchi and Shiota [305]. [1.3] The generalization of the stochastic integral with respect to the Brownian motion to nonadapted processes was introduced by Skorohod in [315], obtaining the isometry formula (1.54), and also by Hitsuda in [136, 135]. The identiﬁcation of the Skorohod integral as the adjoint of the derivative operator has been proved by Gaveau and Trauber [116].

1.5 Sobolev spaces and the equivalence of norms

83

We remark that in [290] (see also Kusuoka [178]) Ramer has also introduced this type of stochastic integral, independently of Skorohod’s work, in connection with the study of nonlinear transformations of the Wiener measure. One can show that the iterated derivative operator Dk is the adjoint of the multiple Skorohod integral δ k , and some of the properties of the Skorohod integral can be extended to multiple integrals (see Nualart and Zakai [264]). Formula (1.63) was ﬁrst proved by Clark [68], where F was assumed to be Frechet ´ diﬀerentiable and to satisfy some technical conditions. In [269] Ocone extends this result to random variables F in the space D1,2 . Clark’s representation theorem has been extended by Karatzas et al. [162] to random variables in the space D1,1 . The spaces L1,2,f and LF of random variables diﬀerentiable in future times were introduced by Alos ` and Nualart in [10]. These spaces lead to a stochastic calculus which generalizes both the classical Itˆo calculus and the Skorohod calculus (see Chapter 3). [1.4] For a complete presentation of the hypercontractivity property and its relation with the Sobolev logarithmic inequality, we refer to the Saint Flour course by Bakry [15]. The multiplier theorem proved in this section is due to Meyer [225], and the proof given here has been taken from Watanabe [343]. [1.5] The Sobolev spaces of Wiener functionals have been studied by diﬀerent authors. In [172] Kr´´ee and Kr´ee proved the continuity of the divergence operator in L2 . k The equivalence between the the norms Dk F p and (−L) 2 p for any p > 1 was ﬁrst established by Meyer [225] using the Littlewood-Payley inequalities. In ﬁnite dimension the operator DC −1 is related to the Riesz transform. Using this idea, Gundy [129] gives a probabilistic proof of Meyer’s inequalities which is based on the properties of the three-dimensional Bessel process and Burkholder inequalities for martingales. On the other hand, using the boundedness in Lp of the Hilbert transform, Pisier [285] provides a short analytical proof of the fact that the operator DC −1 is bounded in Lp . We followed Pisier’s approach in Section 1.5. In [343] Watanabe developed the theory of distributions on the Wiener space that has become a useful tool in the analysis of regularity of probability densities.

2 Regularity of probability laws

In this chapter we apply the techniques of the Malliavin calculus to study the regularity of the probability law of a random vector deﬁned on a Gaussian probability space. We establish some general criteria for the absolute continuity and regularity of the density of such a vector. These general criteria will be applied to the solutions of stochastic diﬀerential equations and stochastic partial diﬀerential equations driven by a space-time white noise.

2.1 Regularity of densities and related topics This section is devoted to study the regularity of the law of a random vector F = (F 1 , . . . , F m ), which is measurable with respect to an underlying isonormal Gaussian process {W (h), h ∈ H}. Using the duality between the operators D and δ we ﬁrst derive an explicit formula for the density of a one-dimensional random variable and we deduce some estimates. Then we establish a criterion for absolute continuity for a random vector under the assumption that its Malliavin matrix is invertible a.s. An alternative approach, due to Bouleau and Hirsch, is presented in the third part of this section. This approach is based on a criterion for absolute continuity in ﬁnite dimension and it then uses a limit argument. The criterion obtained in this way is stronger than that obtained by integration by parts, in that it requires weaker regularity hypotheses on the random vector.

86

2. Regularity of probability laws

We later introduce the notion of smooth and nondegenerate random vector by the condition that the inverse of the determinant of the Malliavin matrix has moments of all orders. We show that smooth and nondegenerate random vectors have inﬁnitely diﬀerentiable densities. Two diﬀerent proofs of this result are given. First we show by a direct argument the local smoothness of the density under more general hypotheses. Secondly, we derive the smoothness of the density from the properties of the composition of a Schwartz tempered distribution with a smooth and nondegenerated random vector. We also study some properties of the topological support of the law of a random vector. The last part of this section is devoted to the regularity of the law of the supremum of a continuous process.

2.1.1 Computation and estimation of probability densities As in the previous chapter, let W = {W (h), h ∈ H} be an isonormal Gaussian process associated to a separable Hilbert space H and deﬁned on a complete probability space (Ω, F, P ). Assume also that F is generated by W . The integration-by-parts formula leads to the following explicit expression for the density of a one-dimensional random variable. Proposition 2.1.1 Let F be a random variable in the space D1,2 . Suppose DF belongs to the domain of the operator δ in L2 (Ω). Then the law that DF

2H of F has a continuous and bounded density given by & ' DF p(x) = E 1{F >x} δ . DF 2H

(2.1)

Proof: Let ψ be a nonnegative smooth function with compact support, y and set ϕ(y) = −∞ ψ(z)dz. We know that ϕ(F ) belongs to D1,2 , and making the scalar product of its derivative with DF obtains D(ϕ(F )), DF H = ψ(F )DF 2H . Using the duality relationship between the operators D and δ (see (1.42)), we obtain . ' DF = E D(ϕ(F )), DF 2H H & ' DF = E ϕ(F )δ . DF 2H &-

E[ψ(F )]

(2.2)

2.1 Regularity of densities and related topics

87

By an approximation argument, Equation (2.2) holds for ψ(y) = 1[a,b] (y), where a < b. As a consequence, we apply Fubini’s theorem to get ( ) F DF ψ(x)dx δ P (a ≤ F ≤ b) = E DF 2H −∞ ' b & DF = E 1{F >x} δ dx, DF 2H a which implies the desired result. We note that suﬃcient conditions for

DF

DF 2H

∈ Dom δ are that F is in

D2,4 and that E(DF −8 H ) < ∞ (see Exercise 2.1.1). On the other hand, DF ∈ Equation (2.1) still holds under the hypotheses F ∈ D1,p and DF

2 H

D1,p (H) for some p, p > 1. We will see later that the property DF H > 0 1,1 ) is suﬃcient for the existence of a density. a.s. (assuming that F is in Dloc From expression (2.1) we can deduce estimates for the density. Fix p and q such that p1 + 1q = 1. By H¨ ¨older’s inequality we obtain

DF 1/q

. δ p(x) ≤ (P (F > x)) 2

DF H p In the same way, taking into account the relation E[δ(DF/DF 2H )] = 0 we can deduce the inequality

DF 1/q

.

p(x) ≤ (P (F < x)) δ 2 DF H p As a consequence, we obtain p(x) ≤ (P (|F | > |x|))

1/q

DF

,

δ 2

DF H p

(2.3)

for all x ∈ R. Now using the Lp (Ω) estimate of the operator δ established in Proposition 1.5.8 we obtain

DF DF DF

≤ cp E

+ D

δ .

DF 2H p DF 2H H DF 2H Lp (Ω;H⊗H) (2.4) We have # 2 $ D F, DF ⊗ DF H⊗H DF D2 F D −2 , = DF 2H DF 2H DF 4H and, hence,

3 D2 F H⊗H

DF

D

≤ .

DF 2H H⊗H DF 2H

(2.5)

88

2. Regularity of probability laws

Finally, from the inequalities (2.3), (2.4) and (2.5) we deduce the following estimate. Proposition 2.1.2 Let q, α, β be three positive real numbers such that 1 1 1 2,α , such that q + α + β = 1. Let F be a random variable in the space D −2β

E(DF H ) < ∞. Then the density p(x) of F can be estimated as follows 1/q

p(x) ≤ cq,α,β (P (|F | > |x|))

2

−1 −2

× E(DF H ) + D F Lα (Ω;H⊗H) DF H . (2.6) β

Let us apply the preceding proposition to a Brownian martingale. Proposition 2.1.3 Let W = {W (t), t ∈ [0, T ]} be a Brownian motion and let u = {u(t), t ∈ [0, T ]} be an adapted process verifying the following hypotheses: T (i) E 0 u(t)2 dt < ∞, u(t) belongs to the space D2,2 for each t ∈ [0, T ], and T p 2 |Dr,s ut |p dt) 2 ) < ∞, λ := sup E(|Ds ut |p ) + sup E(( s,t∈[0,T ]

r,s∈[0,T ]

0

for some p > 3. (ii) |u(t)| ≥ ρ > 0 for some constant ρ. t Set Mt = 0 u(s)dW Ws , and denote by pt (x) the probability density of Mt . Then for any t > 0 we have 1 c Mt | > |x|) q , pt (x) ≤ √ P (|M t

where q >

p p−3

(2.7)

and the constant c depends on λ, ρ and p.

Proof: Fix t ∈ (0, T ]. We will apply Proposition 2.1.2 to the random variable Mt . We claim that Mt ∈ D2,2 . In fact, note ﬁrst that by Lemma 1.3.4 Mt ∈ D1,2 and for s < t t Ds Mt = us + Ds ur dW Wr . (2.8) s

For almost all s, the process {Ds ur , r ∈ [0, T ]} is adapted and belongs to t Wr belongs to D1,2 and L1,2 . Hence, by Lemma 1.3.4 s Ds ur dW Dθ

t

Ds ur dW Wr s

= Ds uθ +

t

s∨θ

Dθ Ds ur dW Wr .

(2.9)

2.1 Regularity of densities and related topics

89

From (2.8) and (2.9) we deduce for any θ, s ≤ t Dθ Ds Mt = Dθ us + Ds uθ +

t

Dθ Ds ur dW Wr .

s∨θ

(2.10)

We will take α = p in Proposition 2.1.2. Using H¨ o¨lder’s and Burkholder’s inequalities we obtain from (2.10)

p

E( D2 Mt H⊗H ) ≤ cp λtp . Set t

2

σ(t) := DM Mt H =

us +

2

t

Ds ur dW Wr

0

ds.

s

We have the following estimates for any h ≤ 1 σ(t) ≥

t(1−h)

≥ ≥

t

2 t Ds ur dW Wr ds us + s

t

u2s ds − 2

t(1−h) 2

t

2

t

Ds ur dW Wr

t(1−h)

ds

s

thρ − Ih (t), 2

where Ih (t) = Choose h =

4 tρ2 y ,

2

t

Ds ur dW Wr

t(1−h)

ds.

s

and notice that h ≤ 1 provided y ≥ a :=

σ(t) ≤

P

t

1 y

≤P

Ih (t) ≥

1 y

p

4 tρ2 .

We have

p

≤ y 2 E(|IIh (t)| 2 ).

(2.11)

Using Burkholder’ inequality for square integrable martingales we get the following estimate p 2

E(|IIh (t)| ) ≤ cp (th)

p 2 −1

t

t

E t(1−h)

(Ds ur ) dr

p2 ds

s

≤ cp sup E(|Ds ur |p )(th)p . s,r∈[0,t]

2

(2.12)

90

2. Regularity of probability laws

Consequently, for 0 < γ y dy 0 ∞ 1 γ γ−1 ≤ a +γ y P σ(t) < dy y a γ ∞ p p 4 +γ E(|IIh (t)| 2 )y γ−1+ 2 dy ≤ 2 4 tρ tρ2 ∞ p −γ γ−1− p 2 dy ≤ c t + y ≤ c t−γ + t 2 −γ .

E(σ(t)−γ ) =

4 tρ2

(2.13)

Substituting (2.13) in Equation (2.6) with α = p, β < p2 , and with γ = 12 and γ = β, we get the desired estimate. Applying Tchebychev and Burkholder’s inequalities, from (2.7) we deduce the following inequality for any θ > 1 c|x|− q pt (x) ≤ √ t

θ

1 θ2 q t 2 us ds . E

0

Corollary 2.1.1 Under the conditions of Proposition 2.1.3, if the process u satisﬁes |ut | ≤ M for some constant M , then c |x|2 pt (x) ≤ √ exp − . qM 2 t t Proof:

It suﬃces to apply the martingale exponential inequality (A.5).

2.1.2 A criterion for absolute continuity based on the integration-by-parts formula We recall that Cb∞ (Rm ) denotes the class of functions f : Rm → R that are bounded and possess bounded derivatives of all orders, and we write ∂ . We start with the following lemma of real analysis (cf. Malliavin ∂i = ∂x i [207]). Lemma 2.1.1 Let µ be a ﬁnite measure on Rm . Assume that for all ϕ ∈ Cb∞ (Rm ) the following inequality holds: ≤ ci ϕ∞ , ∂ ϕdµ 1 ≤ i ≤ m, (2.14) i Rm

where the constants ci do not depend on ϕ. Then µ is absolutely continuous with respect to the Lebesgue measure.

2.1 Regularity of densities and related topics

Proof: If m = 1 there is a simple consider the function ϕ deﬁned by ⎧ ⎨ 0 x−a ϕ(x) = ⎩ b−a 1

91

proof of this result. Fix a < b, and if if if

x≤a a<x
Although this function is not inﬁnitely diﬀerentiable, we can approximate it by functions of Cb∞ (R) in such a way that Eq. (2.14) still holds. In this form we get µ([a, b]) ≤ c1 (b − a), which implies the absolute continuity of µ. For an arbitrary value of m, Malliavin [207] gives a proof of this lemma that uses techniques of harmonic analysis. Following a remark in Malliavin’s paper, we are going to give a diﬀerent proof and show that the density of m µ belongs to L m−1 if m > 1. Consider an approximation of the identity {ψ , > 0} on Rm . Take, for instance, ψ (x) = (2π)− 2 exp(− m

|x|2 ). 2

Let cM (x), M ≥ 1, be a sequence of functions of the space C0∞ (Rm ) such that 0 ≤ cM ≤ 1 and 1 if |x| ≤ M cM (x) = 0 if |x| ≥ M + 1. We assume that the partial derivatives of cM of all orders are bounded uniformly with respect to M . Then the functions ψ (x − y)µ(dy) cM (x)(ψ ∗ µ)(x) = cM (x) Rm

belong to C0∞ (Rm ). The Gagliardo-Nirenberg inequality says that for any function f in the space C0∞ (Rm ) one has m ≤ f L m−1

m

1/m

∂ ∂i f L1 .

i=1

An elementary proof of this inequality can be found in Stein [317, p. 129]. Applying this inequality to the functions cM (ψ ∗ µ), we obtain m ≤ cM (ψ ∗ µ)L m−1

m

1

∂ ∂i (cM (ψ ∗ µ))Lm1 .

(2.15)

i=1

Equation (2.14) implies that the mapping ϕ → ∂ ϕ dµ, deﬁned on Rm i C0∞ (Rm ), is a signed measure, which will be denoted by ν i , 1 ≤ i ≤ m.

92

2. Regularity of probability laws

Then we have

∂i ψ (x − y)µ(dy) dx Rm Rm + |∂ ∂i cM (x)| ψ (x − y)µ(dy) dx Rm Rm dx ψ (x − y)ν (dy) ≤ i Rm Rm + |∂ ∂i cM (x)| ψ (x − y)µ(dy) dx ≤ K,

∂ ∂i (cM (ψ ∗ µ))L1

≤

cM (x)

Rm

Rm

where K is a constant not depending on M and . Consequently, the family m of functions {cM (ψ ∗ µ), M ≥ 1, > 0}mis bounded in L m−1 . We use the weak compactness of the unit ball of L m−1 to deduce the desired result. Suppose that F = (F 1 , . . . , F m ) is a random vector whose components 1,1 . We associate to F the following random symmetric belong to the space Dloc nonnegative deﬁnite matrix: γ F = (DF i , DF j H )1≤i,j≤m . This matrix will be called the Malliavin matrix of the random vector F . The basic condition for the absolute continuity of the law of F will be that the matrix γ F is invertible a.s. The ﬁrst result in this direction follows. Theorem 2.1.1 Let F = (F 1 , . . . , F m ) be a random vector verifying the following conditions: (i) F i ∈ D2,p loc for all i, j = 1, . . . , m, for some p > 1. (ii) The matrix γ F is invertible a.s. Then the law of F is absolutely continuous with respect to the Lebesgue measure on Rm . Proof: We will assume that F i ∈ D2,p for each i. Fix a test function ∞ ϕ ∈ Cb (Rm ). From Proposition 1.2.3, we know that ϕ(F ) belongs to the space D1,p and that D(ϕ(F )) =

m

∂i ϕ(F )DF i .

i=1

Hence, D(ϕ(F )), DF j H =

m

∂i ϕ(F )γ ij F;

i=1

therefore, ∂i ϕ(F ) =

m j=1

ji D(ϕ(F )), DF j H (γ −1 F ) .

(2.16)

2.1 Regularity of densities and related topics

93

The inverse of γ F may not have moments, and for this reason we need a localizing argument. For any integer N ≥ 1 we consider a function ΨN ∈ C0∞ (Rm ⊗ Rm ) such that ΨN ≥ 0 and (a) ΨN (σ) = 1

if σ ∈ KN ,

(b) ΨN (σ) = 0

if σ ∈ / KN +1 , where

KN = {σ ∈ Rm ⊗ Rm : |σ ij | ≤ N

for all

i, j,

Note that KN is a compact subset of GL(m) ⊂ Rm (2.16) by ΨN (γ F ) yields E[ΨN (γ F )∂ ∂i ϕ(F )] =

m

1 }. N ⊗ Rm . Multiplying

and

| det σ| ≥

ji E[ΨN (γ F )D(ϕ(F )), DF j H (γ −1 F ) ].

j=1 ji j 1,p (H). ConseCondition (i) implies that ΨN (γ F )(γ −1 F ) DF belongs to D 1,p quently, we use the continuity of the operator δ from D (H) into Lp (Ω) (Proposition 1.5.4) and the duality relationship (1.42) to obtain

! " ∂i ϕ(F ) E ΨN (γ F )∂

m ji j = E ϕ(F ) δ ΨN (γ F )(γ −1 ) DF F j=1

⎞ ⎛ m ji j ⎠ ≤ E ⎝ δ ΨN (γ F )(γ −1 F ) DF ϕ∞ . j=1 Therefore, by Lemma 2.1.1 the measure [ΨN (γ F ) · P ] ◦ F −1 is absolutely continuous with respect to the Lebesgue measure on Rm . Thus, for any Borel set A ⊂ Rm with zero Lebesgue measure we have ΨN (γ F )dP = 0. F −1 (A)

Letting N tend to inﬁnity and using hypothesis (ii), we obtain the equality P (F −1 (A)) = 0, thereby proving that the probability P ◦ F −1 is absolutely continuous with respect to the Lebesgue measure on Rm . Notice that if we only assume condition (i) in Theorem 2.1.1 and if no nondegeneracy condition on the Malliavin matrix is made, then we deduce that the measure (det(γ F ) · P ) ◦ F −1 is absolutely continuous with respect to the Lebesgue measure on Rm . In other words, the random vector F has an absolutely continuous law conditioned by the set {det(γ F ) > 0}; that is, P {F ∈ B, det(γ F ) > 0} = 0 for any Borel subset B of Rm of zero Lebesgue measure.

94

2. Regularity of probability laws

2.1.3 Absolute continuity using Bouleau and Hirsch’s approach In this section we will present the criterion for absolute continuity obtained by Bouleau and Hirsch [46]. First we introduce some results in ﬁnite dimension, and we refer to Federer [96, pp. 241–245] for the proof of these results. We denote by λn the Lebesgue measure on Rn . Let ϕ be a measurable function from R to R. Then ϕ is said to be approximately diﬀerentiable at a ∈ R, with an approximate derivative equal to b, if lim

η→0

1 1 λ {x ∈ [a − η, a + η] : |ϕ(x) − ϕ(a) − (x − a)b| > |x − a|} = 0 η

for all > 0. We will write b = ap ϕ (a). The following property is an immediate consequence of the above deﬁnition. (a) If ϕ = ϕ ˜ a.e. and ϕ is diﬀerentiable a.e., then ϕ ˜ is approximately diﬀerentiable a.e. and ap ϕ ˜ = ϕ a.e. If ϕ is a measurable function from Rn to R, we will denote by ap ∂i ϕ the approximate partial derivative of ϕ with respect to the ith coordinate. We will also denote by ap ∇ϕ = (ap ∂1 ϕ, . . . , ap ∂n ϕ) the approximate gradient of ϕ. Then we have the following result: Lemma 2.1.2 Let ϕ : Rn → Rm be a measurable function, with m ≤ n, such that the approximate derivatives ap ∂j ϕi , 1 ≤ i ≤ m, 1 ≤ j ≤ n, exist for almost every x ∈ Rn with respect to the Lebesgue measure on Rn . Then we have det[ap ∇ϕj , ap ∇ϕk ]1≤j,k≤m dλn = 0 (2.17) ϕ−1 (B)

for any Borel set B ⊂ Rm with zero Lebesgue measure. Notice that the conclusion of Lemma 2.1.2 is equivalent to saying that (det[ap ∇ϕj , ap ∇ϕk ] · λn ) ◦ ϕ−1 λm . We will also make use of linear transformations of the underlying Gaussian process {W (h), h ∈ H}. Fix an element g ∈ H and consider the translated Gaussian process {W g (h), h ∈ H} deﬁned by W g (h) = W (h) + h, gH . Lemma 2.1.3 The process W g has the same law (that is, the same ﬁnite dimensional distributions) as W under a probability Q equivalent to P given by 1 dQ = exp(−W (g) − g2H ). dP 2

2.1 Regularity of densities and related topics

95

Proof: Let f : Rn → R be a bounded Borel function, and let e1 , . . . , en be orthonormal elements of H. Then we have & ' 1 g g 2 E f (W (e1 ), . . . , W (en )) exp −W (g) − gH 2 g g = E f (W (e1 ), . . . , W (en )) n n 1 2 ei , gH W (ei ) − ei , gH × exp − 2 i=1 i=1 = f (x1 + g, e1 H , . . . , xn + g, en H ) Rn n 1 2 × exp − |xi + g, ei H | dx 2 i=1 = E[f (W (e1 ), . . . , W (en ))]. Now consider a random variable F ∈ L0 (Ω). We can write F = ψ F ◦ W , where ψ F is a measurable mapping from RH to R that is uniquely determined except on a set of measure zero for P ◦ W −1 . By the preceding lemma on the equivalence between the laws of W and W g , we can deﬁne the shifted random variable F g = ψ F ◦W g . Then the following result holds. Lemma 2.1.4 Let F be a random variable in the space D1,p , p > 1. Fix two elements h, g ∈ H. Then there exists a version of the process sh+g , s ∈ R} such that for all a < b we have {DF, hH b sh+g DF, hH ds (2.18) F bh+g − F ah+g = a

a.s. Consequently, there exists a version of the process {F th+g , t ∈ R} that has absolutely continuous paths with respect to the Lebesgue measure on R, . and its derivative is equal to DF, hth+g H Proof:

The proof will be done in two steps.

Step 1: First we will show that F th+g ∈ Lq (Ω) for all q ∈ [1, p) with an Lq norm uniformly bounded with respect to t if t varies in some bounded interval. In fact, let us compute 1 th+g q q 2 E(|F | ) = E |F | exp tW (h) + W (g) − th + gH 2 '1− pq & q p p p ≤ (E(|F | ) (tW (h) + W (g)) E exp p−q 2

1

× e− 2 th+g H =

q

(E(|F |p ) p exp

q th + g2H 2(p − q)

< ∞.

(2.19)

96

2. Regularity of probability laws

Step 2:

Suppose ﬁrst that F is a smooth functional of the form F =

f (W (h1 ), . . . , W (hk )). In this case the mapping t → F th+g is continuously diﬀerentiable and d th+g (F ) dt

k = ∂i f (W (h1 ) + th, h1 H + g, h1 H , i=1

. . . . , W (hk ) + th, hk H + g, hk H )h, hi H = DF, hth+g H Now suppose that F is an arbitrary element in D1,p , and let {F Fk , k ≥ 1} be a sequence of smooth functionals such that as k tends to inﬁnity Fk converges to F in Lp (Ω) and DF Fk converges to DF in Lp (Ω; H). By taking suitable subsequences, we can also assume that these convergences hold almost everywhere. We know that for any k and any a < b we have b sh+g Fkbh+g − Fkah+g = DF Fk , hH ds. (2.20) a

Fkth+g

For any t ∈ R the random variables converge almost surely to F th+g as k tends to inﬁnity. On the other hand, the sequence of random variables b b sh+g sh+g DF Fk , hH ds converges in L1 (Ω) to a DF, hH ds as k tends to a inﬁnity. In fact, using Eq. (2.19) with q = 1, we obtain b b sh+g sh+g E DF Fk , hH ds − DF, hH ds a a b sh+g sh+g ≤ E |DF Fk , hH − DF, hH |ds a

≤

1 E(|Dh Fk − Dh F |p ) p (b − a) 1 th + g2H . × sup exp 2(p − 1) t∈[a,b]

In conclusion, by taking the limit of both sides of Eq. (2.20) as k tends to inﬁnity, we obtain (2.18). This completes the proof. Here is a useful consequence of Lemma 2.1.4. Lemma 2.1.5 Let F be a random variable in the space D1,p for some p > 1. Fix h ∈ H. Then, a.s. we have 1 th lim (F − F )dt = DF, hH . (2.21) →0 0 Proof:

By Lemma 2.1.4, for almost all (ω, x) ∈ Ω × R we have 1 x+ yh lim (F (ω) − F (ω))dy = DF (ω), hxh H . →0 x

(2.22)

2.1 Regularity of densities and related topics

97

Hence, there exists an x ∈ R for which (2.22) holds a.s. Finally, if we consider the probability Q deﬁned by dQ x2 = exp(−xW (h) − h2H ) dP 2 we obtain that (2.21) holds Q a.s. This completes the proof.

Now we can state the main result of this section. Theorem 2.1.2 Let F = (F 1 , . . . , F m ) be a random vector satisfying the following conditions: (i) F i belongs to the space D1,p loc , p > 1, for all i = 1, . . . , m. (ii) The matrix γ F = (DF i , DF j )1≤i,j≤m is invertible a.s. Then the law of F is absolutely continuous with respect to the Lebesgue measure on Rm . Proof: We may assume by a localization argument that F k belongs to 1,p for k = 1, . . . , m. Fix a complete orthonormal system {ei , i ≥ 1} in D the Hilbert space H. For any natural number n ≥ 1 we deﬁne ϕn,k (t1 , . . . , tn ) = (F k )t1 e1 +···+tn en , for 1 ≤ k ≤ m. By Lemma 2.1.4, if we ﬁx the coordinates t1 , . . . , ti−1 , ti+1 , . . . , tn , the process {ϕn,k (t1 , . . . , tn ), ti ∈ R} has a version with absolutely continuous paths. So, for almost all t the function ϕn,k (t1 , . . . , tn ) has an approximate partial derivative with respect to the ith coordinate, and moreover, ap∂ ∂i ϕn,k (t) = DF k , ei tH1 e1 +···+tn en . Consequently, we have n ap∇ϕn,k , ap∇ϕn,j = ( DF k , ei H DF j , ei H )t1 e1 +···+tn en .

(2.23)

i=1

Let B be a Borel subset of Rm of zero Lebesgue measure. Then, Lemma 2.1.2 applied to the function ϕn = (ϕn,1 , . . . , ϕn,m ) yields, for almost all ω, assuming n ≥ m (ϕn )−1 (B)

det[ap∇ϕn,k , ap∇ϕn,j ]dt1 . . . dtn = 0.

98

2. Regularity of probability laws

Set G = {t ∈ Rn : F t1 e1 +···+tn en (ω) ∈ B}. Taking expectations in the above expression and using (2.23), we deduce t1 e1 +···+tn en n k j dt1 · · · dtn det( DF , ei H DF , ei H ) 0 = E G

2

=

E Rn

i=1 n

det(

DF k , ei H DF j , ei H )1F −1 (B)

i=1

3 n 1 2 × exp( (ti W (ei ) − ti )) dt1 · · · dtn . 2 i=1 Consequently, n 1F −1 (B) det( DF k , ei H DF j , ei H ) = 0 i=1

almost surely, and letting n tend to inﬁnity yields 1F −1 (B) det(DF k DF j H ) = 0, almost surely. Therefore, P (F −1 (B)) = 0, and the proof of the theorem is complete. As in the remark after the proof of Theorem 2.1.1, if we only assume condition (i) in Theorem 2.1.2, then the measure (det(DF k , DF j H ) · P ) ◦ F −1 is absolutely continuous with respect to the Lebesgue measure on Rm . The following result is a version of Theorem 2.1.2 for one-dimensional random variables. The proof we present here, which has been taken from [266], is much shorter than the proof of Theorem 2.1.2. It even works for p = 1. 1,1 , and suppose Theorem 2.1.3 Let F be a random variable of the space Dloc that DF H > 0 a.s. Then the law of F is absolutely continuous with respect to the Lebesgue measure on R.

Proof: By the standard localization argument we may assume that F belongs to the space D1,1 . Also, we can assume that |F | < 1. We have to show that for any measurable function g : (−1, 1) → [0, 1] such that 1 g(y)dy = 0 we have E(g(F )) = 0. We can ﬁnd a sequence of continu−1 ously diﬀerentiable functions with bounded derivatives g n : (−1, 1) → [0, 1] such that as n tends to inﬁnity g n (y) converges to g(y) for almost all y with respect to the measure P ◦ F −1 + λ1 . Set y ψ n (y) = g n (x)dx −1

2.1 Regularity of densities and related topics

and

99

y

g(x)dx.

ψ(y) = −1

By the chain rule, ψ n (F ) belongs to the space D1,1 and we have D[ψ n (F )] = g n (F )DF . We have that ψ n (F ) converges to ψ(F ) a.s. as n tends to inﬁnity, because g n converges to g a.e. with respect to the Lebesgue measure. This convergence also holds in L1 (Ω) by dominated convergence. On the other hand, Dψ n (F ) converges a.s. to g(F )DF because g n converges to g a.e. with respect to the law of F . Again by dominated convergence, this convergence holds in L1 (Ω; H). Observe that ψ(F ) = 0 a.s. Now we use the property that the operator D is closed to deduce that g(F )DF = 0 a.s. Consequently, g(F ) = 0 a.s., which completes the proof of the theorem. As in the case of Theorems 2.1.1 and 2.1.2, the proof of Theorem 2.1.3 yields the following result: 1,1 . Then the measure Corollary 2.1.2 Let F be a random variable in Dloc −1 is absolutely continuous with respect to the Lebesgue (DF H · P ) ◦ F measure.

This is equivalent to saying that the random variable F has an absolutely continuous law conditioned by the set {DF H > 0}; this means that P {F ∈ B, DF H > 0} = 0 for any Borel subset of R of zero Lebesgue measure.

2.1.4 Smoothness of densities In order to derive the smoothness of the density of a random vector we will impose the nondegeneracy condition given in the following deﬁnition. Deﬁnition 2.1.1 We will say that a random vector F = (F 1 , . . . , F m ) whose components are in D∞ is nondegenerate if the Malliavin matrix γ F is invertible a.s. and −1

(det γ F )

∈ ∩p≥1 Lp (Ω).

We aim to cover some examples of random vectors whose components are not in D∞ and satisfy a local nondegenerary condition. In these examples, the density of the random vector will be smooth only on an open subset of Rm . To handle these example we introduce the following deﬁnition. Deﬁnition 2.1.2 We will say that a random vector F = (F 1 , . . . , F m ) whose components are in D1,2 is locally nondegenerate in an open set A ⊂ Rm if there exist elements ujA ∈ D∞ (H), j = 1, . . . , m and an m × m ij ∞ −1 random matrix γ A = (γ ij ∈ Lp (Ω) for all A ) such that γ A ∈ D , | det γ A | j ij i p ≥ 1, and DF , uA H = γ A on {F ∈ A} for any i, j = 1, . . . , m.

100

2. Regularity of probability laws

Clearly, a nondegenerate random vector is also locally nondegenerate in Rm , and we can take ujRm = DF j , and γ A = γ F . We need the following preliminary lemma. Lemma 2.1.6 Suppose that γ is an m×m random matrix that is invertible a.s. and such that | det γ|−1 ∈ Lp (Ω) for all p ≥ 1. Suppose that the entries ij γ ij of γ are in D∞ . Then γ −1 belongs to D∞ for all i, j, and m ij −1 ik −1 lj D γ −1 = − γ γ Dγ kl .

(2.24)

k,l=1

Proof: First notice that {det γ > 0} has probability zero or one (see Exercise 1.3.4). We will assume that det γ > 0 a.s. For any > 0 deﬁne γ −1 =

det γ γ −1 . det γ +

Note that (det γ + )−1 belongs to D∞ because it can be expressed as the composition of det γ with a function in Cp∞ (R). Therefore, the entries of −1 ij ∞ converges in Lp (Ω) to γ −1 belong to D . Furthermore, for any i, j, γ −1 ij γ as tends to zero. Then, in order to check that the entries of γ −1 belong to D∞ , it suﬃces to show (taking into account Lemma 1.5.3) that ij the iterated derivatives of γ −1 are bounded in Lp (Ω), uniformly with respect to , for any p ≥ 1. This boundedness in Lp (Ω) holds, from the Leibnitz rule for the operator Dk (see Exercise 1.2.13), because (det γ)γ −1 belongs to D∞ , and on the other hand, (det γ + )−1 has bounded · k,p norms for all k, p, due to our hypotheses. det γ Finally, from the expression γ −1 γ = det γ+ I, we deduce Eq. (2.24) by ﬁrst applying the derivative operator D and then letting tend to zero. For a locally nondegenerate random vector the following integration-byparts formula plays a basic role. For any multiindex α ∈ {1, . . . , m}k , k ≥ 1 k we will denote by ∂α the partial derivative ∂xα ∂···∂xα . 1

1

k

m

Proposition 2.1.4 Let F = (F , . . . , F ) be a locally nondegenerate random vector in an open set A ⊂ Rm in the sense of Deﬁnition 2.1.2. Let G ∈ D∞ and let ϕ be a function in the space Cp∞ (Rm ). Suppose that G = 0 on the set {F ∈ / A}. Then for any multiindex α ∈ {1, . . . , m}k , k ≥ 1, there exists an element Hα ∈ D∞ such that Hα ] . E [∂α ϕ(F )G] = E [ϕ(F )H

(2.25)

Moreover, the elements Hα are recursively given by H(i)

=

m ij j δ G γ −1 uA , A

(2.26)

j=1

Hα

= Hαk (H(α1 ,...,αk−1 ) ),

(2.27)

2.1 Regularity of densities and related topics

101

and for 1 ≤ p < q < ∞ we have

k

H Hα p ≤ cp,q γ −1 A u k,2k−1 r Gk,q , where Proof:

1 p

=

1 q

(2.28)

+ 1r .

By the chain rule (Proposition 1.2.3) we have on {F ∈ A} D(ϕ(F )), ujA H

=

m

∂i ϕ(F )DF

i=1

i

, ujA H

=

m

∂i ϕ(F )γ ij A,

i=1

and, consequently, ∂i ϕ(F ) =

m

ji D(ϕ(F )), ujA H (γ −1 A ) .

j=1

Taking into account that G vanishes on the set {F ∈ A}, we obtain G∂ ∂i ϕ(F ) =

m

ji GD(ϕ(F )), ujA H (γ −1 A ) .

j=1

Finally, taking expectations and using the duality relationship between the derivative and the divergence operators we get " ! E [∂ ∂i ϕ(F )G] = E ϕ(F )H(i) , where H(i) equals to the right-hand side of Equation (2.26). Equation (2.27) follows by recurrence. Using the continuity of the operator δ from D1,p (H) into Lp (Ω) and the H¨ older inequality for the ·p,k norms (Proposition 1.5.6) we obtain H Hα p

m

−1 αk j j

γA ≤ cp H(α1 ,...,αk−1 ) u

j=1 1,p

−1 αk

≤ cp H(α1 ,...,αk−1 ) 1,q γ A u . 1,r

This implies (2.28) for k = 1, and the general case follows by recurrence. If F is nondegenerate then Equation (2.25) holds for any G ∈ D∞ , and we replace in this equation γ A and ujA by γ F and DF j , respectively. In that case, the element Hα depends only on F and G and we denote it by Hα (F, G). Then, formulas (2.25) to (2.28) are tranformed into E [∂α ϕ(F )G] = E [ϕ(F )H Hα (F, G)] ,

(2.29)

102

2. Regularity of probability laws

where H(i) (F, G)

=

m ij δ G γ −1 DF j , F

(2.30)

j=1

Hα (F, G) and

= Hαk (H(α1 ,...,αk−1 ) (F, G)),

k

H Hα (F, G)p ≤ cp,q γ −1 F DF k,2k−1 r Gk,q .

(2.31) (2.32)

As a consequence, there exists constants β, γ > 1 and integers n, m such that

m

DF n G . H Hα (F, G)p ≤ cp,q det γ −1 F k,γ k,q β Now we can state the local criterion for smoothness of densities which allows us to show the smoothness of the density for random variables that are not necessarily in the space D∞ . Theorem 2.1.4 Let F = (F 1 , . . . , F m ) be a locally nondegenerate random vector in an open set A ⊂ Rm in the sense of Deﬁnition 2.1.2. Then F possesses an inﬁnitely diﬀerentiable density on the open set A. Proof: Fix x0 ∈ A, and consider an open ball Bδ (x0 ) of radius δ < 1 c d(x , A ). Let δ < δ < d(x0 , Ac ). Consider a function ψ ∈ C ∞ (Rm ) such 0 2 that 0 ≤ ψ(x) ≤ 1, ψ(x) = 1 on Bδ (x0 ), and ψ(x) = 0 on the complement of Bδ (x0 ). Equality (2.25) applied to the multiindex α = (1, 2, . . . , m) and to the random variable G = ψ(F ) yields, for any function ϕ in Cp∞ (Rm ) E [ψ(F )∂α ϕ(F )] = E[ϕ(F )H Hα ]. Notice that

F1

ϕ(F ) = −∞

···

Fm

−∞

∂α ϕ(x)dx.

Hence, by Fubini’s theorem we can write " ! E [ψ(F )∂α ϕ(F )] = ∂α ϕ(x)E 1{F >x} Hα dx.

(2.33)

Rm

We can take as ∂α ϕ any function in C0∞ (Rm ). Then Equation (2.33) implies that on the ball Bδ (x0 ) the random vector F has a density given by " ! p(x) = E 1{F >x} Hα . Moreover, for any multiindex β we have E [ψ(F )∂ ∂β ∂α ϕ(F )]

= E[ϕ(F )H Hβ (H Hα )] ! " = ∂α ϕ(x)E 1{F >x} Hβ (H Hα ) dx. Rm

2.1 Regularity of densities and related topics

103

Hence, for any ξ ∈ C0∞ (Bδ (x0 ))

Rm

∂β ξ(x)p(x)dx =

Rm

! " ξ(x)E 1{F >x} Hβ (H Hα ) dx.

Therefore p(x) is inﬁnitely diﬀerentiable in the ball Bδ (x0 ), and for any multiindex β we have ! " Hα ) . ∂β p(x) = (−1)|β| E 1{F >x} Hβ (H We denote by S(Rm ) the space of all inﬁnitely diﬀerentiable functions f : Rm → R such that for any k ≥ 1, and for any multiindex β ∈ {1, . . . , m}j one has supx∈Rm |x|k |∂ ∂β f (x)| < ∞ (Schwartz space). Proposition 2.1.5 Let F = (F 1 , . . . , F m ) be a nondegenerate random vector in the sense of Deﬁnition 2.1.1. Then the density of F belongs to the space S(Rm ), and " ! p(x) = E 1{F >x} H(1,2,...,m) (F, 1) .

(2.34)

Proof: The proof of Theorem 2.1.4 implies, taking G = 1, that F possesses an inﬁnitely diﬀerentiable density and (2.34) holds. Moreover, for any multiindex β ! " ∂β p(x) = (−1)|β| E 1{F >x} Hβ (H(1,2,...,m) (F, 1)) . In order to show that the density belongs to S(Rm ) we have to prove that for any multiindex β and for any k ≥ 1 and for all j = 1, . . . , m ! " sup x2k j |E 1{F >x} Hβ (H(1,2,...,m) (F, 1)) | < ∞.

x∈Rm

If xj > 0 we have ! " x2k j |E 1{F >x} Hβ (H(1,2,...,m) (F, 1)) | " ! Hβ (H(1,2,...,m) (F, 1))| < ∞. ≤ E |F j |2k |H If xj < 0 then we use the alternative expression for the density ⎡ ⎤ p(x) = E ⎣ 1{xi F j } H(1,2,...,m) (F, 1)⎦ , i= j

and we deduce a similar estimate.

104

2. Regularity of probability laws

2.1.5 Composition of tempered distributions with nondegenerate random vectors Let F be an m-dimensional random vector. The probability density of F at x ∈ Rm can be formally deﬁned as the generalized expectation E(δ x (F )), where δ x denotes the Dirac function at x. The expression E(δ x (F )) can be interpreted as the coupling δ x (F ), 1, provided we show that δ x (F ) is an element of D−∞ . The Dirac function δ x is a measure, and more generally we will see that we can deﬁne the composition T (F ) of a Schwartz distribution T ∈ S (Rm ) with a nondegenerate random vector, and the composition will belong to D−∞ . Furthermore, the diferentiability of the mapping x → δ x (F ) from Rm into some Sobolev space D−k,p provides an alternative proof of the smoothness of the density of F . Consider the following sequence of seminorms in the space S(Rm ):

φ2k = (1 + |x|2 − ∆)k φ ∞ , φ ∈ S(Rm ), (2.35) for k ∈ Z. Let S2k , k ∈ Z, be the completion of S(Rm ) by the seminorm ·2k . Then we have S2k+2 ⊂ S2k ⊂ · · · ⊂ S2 ⊂ S0 ⊂ S−2 ⊂ · · · ⊂ S−2k ⊂ S−2k−2 , m ) is the space of continuous functions on Rm which vanish and S0 = C(R at inﬁnity. Moreover, ∩k≥1 S2k = S(Rm ) and ∪k≥1 S−2k = S (Rm ). Proposition 2.1.6 Let F = (F 1 , . . . , F m ) be a nondegenerate random vector in the sense of Deﬁnition 2.1.1. For any k ∈ N and p > 1, there exists a constant c(p, k, F ) such that for any φ ∈ S(Rm ) we have φ(F )−2k,p ≤ c(p, k, F ) φ−2k . Proof: Let ψ = (1 + |x|2 − ∆)−k φ ∈ S(Rm ). By Proposition 2.1.4 for any G ∈ D∞ there exists R2k (G) ∈ D∞ such that ! " E [φ(F )G] = E (1 + |x|2 − ∆)k ψ(F )G = E [ψ(F )R2k (G)] . Therefore, using (2.35) and (2.28) with q such that

1 p

+

1 q

= 1, yields

|E [φ(F )G]| ≤ ψ∞ E [|R2k (G)|] ≤ c(p, k, F ) φ−2k G2k,q . Finally, it suﬃces to use the fact that 4 5 |φ(F )|−2k,p = sup |E [φ(F )G]| , G ∈ D2k,q , |G|2k,q ≤ 1 . Corollary 2.1.3 Let F be a nondegenerate random vector. For any k ∈ N and p > 1 we can uniquely extend the mapping φ → φ(F ) to a continuous linear mapping from S−2k into D−2k,p .

2.1 Regularity of densities and related topics

105

As a consequence of the above Corollary, we can deﬁne the composition of a Schwartz distribution T ∈ S (Rm ) with the nondegenerate random vector F , as a generalized random variable T (F ) ∈ D−∞ . Actually, −2k,p . T (F ) ∈ ∪∞ k=1 ∩p>1 D

For k = 0, φ(F ) coincides with the usual composition of the continuous m ) and the random vector F . function φ ∈ S0 = C(R m " any x ∈ R , the Dirac function δ x belongs to S−2k , where k = !mFor 2 + 1, and the mapping x → δ x is 2j continuously diﬀerentiable from Rm to S−2k−2j , for any j ∈ N. Therefore, for any nondegenerate random vector F , the composition δ x (F ) belongs to D−2k,p for any p > 1, and the mapping x → δ x (F ) is 2j continuously diﬀerentiable from Rm to D−2k−2j,p , for any j ∈ N. This implies that for any G ∈ D2k+2j,p the mapping x → δ x (F ), G belongs to C 2j (Rm ). ! " m Lemma 2.1.7 Let k = m 2 + 1 and p > 1. If f ∈ C0 (R ), then for any 2k,q G∈D f (x) δ x (F ), G dx = E [f (F )G] . Rm

Proof:

We have

f= Rm

f (x)δ x dx,

where the integral is S−2k -valued and in the sense of Bochner. Thus, approximating the integral by Riemann sums we obtain f (x)δ x (F )dx, f (F ) = Rm

in D−2k,p . Finally, multiplying by G and taking expectations we get the result. This lemma and previous remarks imply that for any G ∈ D2k+2j,p , the measure ! " µG (B) = E 1{F ∈B} G , B ∈ B(Rm ) has a density pG (x) = δ x (F ), G ∈ C 2j (Rm ). In particular, δ x (F ), 1 is the density of F and it will be inﬁnitely diﬀerentiable.

2.1.6

Properties of the support of the law

Given a random vector F : Ω → Rm , the topological support of the law of F is deﬁned as the set of points x ∈ Rm such that P (|x − F | < ε) > 0 for all ε > 0. The following result asserts the connectivity property of the support of a smooth random vector.

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2. Regularity of probability laws

Proposition 2.1.7 Let F = (F 1 , . . . , F m ) be a random vector whose components belong to D1,p for some p > 1. Then, the topological support of the law of F is a closed connected subset of Rm . Proof: If the support of F is not connected, it can be decomposed as the union of two nonempty disjoint closed sets A and B. For each integer M ≥ 2 let ψ M : Rm → R be an inﬁnitely diﬀerentiable function such that 0 ≤ ψ M ≤ 1, ψ M (x) = 0 if |x| ≥ M , ψ M (x) = 1 if |x| ≤ M − 1, and supx,M |∇ψ M (x)| < ∞. Set AM = A ∩ {|x| ≤ M } and BM = B ∩ {|x| ≤ M }. For M large enough we have AM = ∅ and BM = ∅, and there exists an inﬁnitely diﬀerentiable function fM such that 0 ≤ fM ≤ 1, fM = 1 in a neighborhood of AM , and fM = 0 in a neighborhood of BM . The sequence (ffM ψ M )(F ) converges a.s. and in Lp (Ω) to 1{F ∈A} as M tends to inﬁnity. On the other hand, we have D [(ffM ψ M )(F )]

=

m ! " (ψ M ∂i fM )(F )DF i + (ffM ∂i ψ M )(F )DF i i=1

=

m

(ffM ∂i ψ M )(F )DF i .

i=1

Hence, sup D [(ffM ψ M )(F )]H ≤ M

m

sup ∂ ∂i ψ M ∞ DF i H ∈ Lp (Ω).

i=1 M

By Lemma 1.5.3 we get that 1{F ∈A} belongs to D1,p , and by Proposition 1.2.6 this is contradictory because 0 1 is a closed interval. The next result provides suﬃcient conditions for the density of F to be nonzero in the interior of the support. Proposition 2.1.8 Let F ∈ D1,p , p > 2, and suppose that F possesses a density p(x) which is locally Lipschitz in the interior of the support of the law of F . Let a be a point in the interior of the support of the law of F . Then p(a) > 0. 2p > 1. From Proposition 1.2.6 we Proof: Suppose p(a) = 0. Set r = p+2 1,r know that 1{F >a} ∈ D because 0 a) < 1. Fix > 0 and set x 1 1[a−,a+] (y)dy. ϕ (x) = 2 −∞

Then ϕ (F ) converges to 1{F >a} in Lr (Ω) as ↓ 0. Moreover, ϕ (F ) ∈ D1,r and 1 D(ϕ (F )) = 1[a−,a+] (F )DF. 2

2.1 Regularity of densities and related topics

107

We have E

(D(ϕ (F ))rH )

≤

2 (E(DF pH ) p+2

1 (2)2

p p+2

a+

p(x)dx

.

a−

The local Lipschitz property of p implies that p(x) ≤ K|x − a|, and we obtain 2 p E (D(ϕ (F ))rH ) ≤ (E(DF pH ) p+2 2−r K p+2 . By Lemma 1.5.3 this implies 1{F >a} ∈ D1,r , resulting in a contradiction. Suﬃcient conditions for the density of F to be continuously diﬀerentiable are given in Exercise 2.1.8. The following example shows that, unlike the one-dimensional case, in dimension m > 1 the density of a nondegenerate random vector may vanish in the interior of the support. Example 2.1.1 Let h1 and h2 be two orthonormal elements of H. Deﬁne X = (X1 , X2 ), where X1

= arctan W (h1 ),

X2

= arctan W (h2 ).

Then, Xi ∈ D∞ and DXi = (1 + W (hi )2 )−1 hi , for i = 1, 2, and ! "−2 . det γ X = (1 + W (h1 )2 )(1 + W (h2 )2 ) ! "2 The support of the law of the random vector X is the rectangle − π2 , π2 , and the density of X is strictly positive in the interior of the support. Now consider the vector Y = (Y Y1 , Y2 ) given by Y1

=

Y2

=

3π ) cos(2X2 + π), 2 3π ) sin(2X2 + π). (X1 + 2 (X1 +

We have that Yi ∈ D∞ for i = 1, 2, and det γ Y = 4(X1 +

"−2 3π 2 ! ) (1 + W (h1 )2 )(1 + W (h2 )2 ) . 2

This implies that Y is a nondegenerate random vector. Its support is the set {(x, y) : π 2 ≤ x2 + y 2 ≤ 4π 2 }, and the density of Y vanishes on the points (x, y) in the support such that π < y < 2π and x = 0.

108

2. Regularity of probability laws

For a nondegenerate random vector when the density vanishes, then all its partial derivatives also vanish. Proposition 2.1.9 Let F = (F 1 , . . . , F m ) be a nondegenerate random vector in the sense of Deﬁnition 2.1.1 and denote its density by p(x). Then p(x) = 0 implies δ α p(x) = 0 for any multiindex α. Proof: Suppose that p(x) = 0. For any nonnegative random variable G !∈ D∞ , δ x"(F ), G ≥ 0 because this is the density of the measure µG (B) = E 1{F ∈B} G , B ∈ B(Rm ). Fix a complete orthonormal system {ei , i ≥ 1} in H. For each n ≥ 1 the function ϕ : Rn → C given by ⎛ ⎞7 6 n ϕ(t) = δ x (F ), exp ⎝i tj W (ej )⎠ j=1

is nonnegative deﬁnite and continuous. Thus, there exists a measure ν n on Rn such that eit,x dν n (x). ϕ(t) = Rn

Note that ν n (R ) = δ x (F ), 1 = p(x) = 0. So, this measure is zero and we get that δ x (F ), G = 0 for any polynomial random variable G ∈ P. This implies that δ x (F ) = 0 as an element of D−∞ . For any multiindex α we have n

∂α p(x) = ∂α δ x (F ), 1 = (∂α δ x ) (F ), 1 . Hence, it suﬃces to show that (∂α δ x ) (F ) vanishes. Suppose ﬁrst that α = {i}. We can write D (δ x (F )) =

m

(∂ ∂i δ x ) (F )DF i

i=1

as elements of D−∞ , which implies (∂ ∂i δ x ) (F ) =

m #

D (δ x (F )) , DF j

$ H

ji (γ −1 F ) =0

j=1

because D (δ x (F )) = 0. The general case follows by recurrence.

2.1.7

Regularity of the law of the maximum of continuous processes

In this section we present the application of the Malliavin calculus to the absolute continuity and smoothness of the density for the supremum of a continuous process. We assume that the σ-algebra of the underlying

2.1 Regularity of densities and related topics

109

probability space (Ω, F, P ) is generated by an isonormal Gaussian process W = {W (h), h ∈ H}. Our ﬁrst result provides suﬃcient conditions for the diﬀerentiability of the supremum of a continuous process. Proposition 2.1.10 Let X = {X(t), t ∈ S} be a continuous process parametrized by a compact metric space S. Suppose that (i) E(supt∈S X(t)2 ) < ∞; (ii) for any t ∈ S, X(t) ∈ D1,2 , the H-valued process {DX(t), t ∈ S} possesses a continuous version, and E(supt∈S DX(t)2H ) < ∞. Then the random variable M = supt∈S X(t) belongs to D1,2 . Proof: Consider a countable and dense subset S0 = {tn , n ≥ 1} in S. Deﬁne Mn = sup{X(t1 ), . . . , X(tn )}. The function ϕn : Rn → R deﬁned by ϕn (x1 , . . . , xn ) = max{x1 , . . . , xn } is Lipschitz. Therefore, from Proposition 1.2.4 we deduce that Mn belongs to D1,2 . The sequence Mn converges in L2 (Ω) to M . Thus, by Lemma 1.2.3 it suﬃces to see that the sequence DM Mn is bounded in L2 (Ω; H). In order to evaluate the derivative of Mn , we introduce the following sets: Mn = X(t1 )}, A1 = {M ··· Ak = {M Mn = X(t1 ), . . . , Mn = X(tk−1 ), Mn = X(tk )},

2 ≤ k ≤ n.

By the local property of the operator D, on the set Ak the derivatives of the random variables Mn and X(tk ) coincide. Hence, we can write DM Mn =

n

1Ak DX(tk ).

k=1

Consequently, E(DM Mn 2H ) and the proof is complete.

≤E

sup DX(t)2H t∈S

< ∞,

We can now establish the following general criterion of absolute continuity. Proposition 2.1.11 Let X = {X(t), t ∈ S} be a continuous process parametrized by a compact metric space S verifying the hypotheses of Proposition 2.1.10. Suppose that DX(t)H = 0 on the set {t : X(t) = M }. Then the law of M = supt∈S X(t) is absolutely continuous with respect to the Lebesgue measure.

110

2. Regularity of probability laws

Proof: By Theorem 2.1.3 it suﬃces to show that a.s. DM = DX(t) on the set {t : X(t) = M }. Thus, if we deﬁne the set G = {there exists t ∈ S : DX(t) = DM, and X(t) = M }, then P (G) = 0. Let S0 = {tn , n ≥ 1} be a countable and dense subset of S. Let H0 be a countable and dense subset of the unit ball of H. We can write 8 G⊂ Gs,r,k,h , s∈S0 ,r∈Q,r>0,k≥1,h∈H0

where Gs,r,k,h = {DX(t) − DM, hH >

1 for all t ∈ Br (s)} ∩ { sup Xt = M }. k t∈Br (s)

Here Br (s) denotes the open ball with center s and radius r. Because it is a countable union, it suﬃces to check that P (Gs,r,k,h ) = 0 for ﬁxed s, r, k, h. Set M = sup{X(t), t ∈ Br (s)} and Mn = sup{X(ti ), 1 ≤ i ≤ n, ti ∈ Br (s)}. By Lemma 1.2.3, DM Mn converges to DM in the weak topology of 2 L (Ω; H) as n tends to inﬁnity, but on the set Gs,r,k,h we have DM Mn − DM , hH ≥ for all n ≥ 1. This implies that P (Gs,r,k,h ) = 0.

1 k

Consider the case of a continuous Gaussian process X = {X(t), t ∈ S} with covariance function K(s, t), and suppose that the Gaussian space H1 is the closed span of the random variables X(t). We can choose as Hilbert space H the closed span of the functions {K(t, ·), t ∈ S} with the scalar product K(t, ·), K(s, ·)H = K(t, s), that is, H is the reproducing kernel Hilbert space (RKHS) (see [13]) associated with the process X. The space H contains all functions of the form ϕ(t) = E(Y X(t)), where Y ∈ H1 . Then, DX(t) = K(t, ·) and DX(t)H = K(t, t). As a consequence, the criterion of the above proposition reduces to K(t, t) = 0 on the set {t : X(t) = M }. Let us now discuss the diﬀerentiability of the density of M = supt∈S X(t). If S = [0, 1] and the process X is a Brownian motion, then the law of M has the density x2 2 p(x) = √ e− 2 1[0,∞) (x). 2π Indeed, the reﬂection principle (see [292, Proposition III.3.7]) implies that P {supt∈[0,1] X(t) > a} = 2P {X(1) > a} for all a > 0. Note that p(x) is inﬁnitely diﬀerentiable in (0, +∞).

2.1 Regularity of densities and related topics

111

Consider now the case of a two-parameter Wiener process on the unit square W = {W (z), z ∈ [0, 1]2 }. That is, S = T = [0, 1]2 and µ is the Lebesgue measure. Set M = supz∈[0,1]2 W (z). The explicit form of the density of M is unknown. We will show that the density of M is inﬁnitely diﬀerentiable in (0, +∞), but ﬁrst we will show some preliminary results. Lemma 2.1.8 With probability one the Wiener sheet W attains its maximum on [0, 1]2 on a unique random point (S, T ). Proof: G=

We want to show that the set 2

3

ω : sup W (z) = W (z1 ) = W (z2 ) for

some

z∈[0,1]2

z1 = z2

has probability zero. For each n ≥ 1 we denote by Rn the class of dyadic rectangles of the form [(j − 1)2−n , j2−n ] × [(k − 1)2−n , k2−n ], with 1 ≤ j, k ≤ 2n . The set G is included in the countable union 8 8 sup W (z) = sup W (z) . n≥1 R1 ,R2 ∈Rn ,R1 ∩R2 =∅

z∈R1

z∈R2

Finally, it suﬃces to check that for each n ≥ 1 and for any couple of disjoint rectangles R1 , R2 with sides parallel to the axes, P {supz∈R1 W (z) = supz∈R2 W (z)} = 0 (see Exercise 2.1.7). Lemma 2.1.9 The random variable M = supz∈[0,1]2 W (z) belongs to D1,2 and Dz M = 1[0,S]×[0,T ] (z), where (S, T ) is the point where the maximum is attained. Proof:

We introduce the approximation of M deﬁned by Mn = sup{W (z1 ), . . . , W (zn )},

where {zn , n ≥ 1} is a countable and dense subset of [0, 1]2 . It holds that Dz Mn = 1[0,Sn ]×[0,TTn ] (z), where (Sn , Tn ) is the point where Mn = W (Sn , Tn ). We know that the sequence of derivatives DM Mn converges to DM in the weak topology of L2 ([0, 1]2 × Ω). On the other hand, (Sn , Tn ) converges to (S, T ) almost surely. This implies the result. As an application of Theorem 2.1.4 we can prove the regularity of the density of M . Proposition 2.1.12 The random variable M = supz∈[0,1]2 W (z) possesses an inﬁnitely diﬀerentiable density on (0, +∞).

112

2. Regularity of probability laws

Proof: Fix a > 0 and set A = (a, +∞). By Theorem 2.1.4 it suﬃces to show that M is locally nondegenerate in A in the sense of Deﬁnition 2.1.2. Deﬁne the following random variables: Ta = inf{t :

sup

W (x, y) > a}

{0≤x≤1,0≤y≤t}

and Sa = inf{s :

sup

W (x, y) > a}.

{0≤x≤s,0≤y≤1}

We recall that Sa and Ta are stopping times with respect to the oneparameter ﬁltrations Fs1 = σ{W (x, y) : 0 ≤ x ≤ s, 0 ≤ y ≤ 1} and Ft2 = σ{W (x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ t}. Note that (Sa , Ta ) ≤ (S, T ) on the set {M > a}. Hence, by Lemma 2.1.9 it holds that Dz M (ω) = 1 for almost all (z, ω) such that z ≤ (Sa (ω), Ta (ω)) and M (ω) > a. For every 0 < γ < 12 and p > 2 such that Holder ¨ seminorm on C0 ([0, 1]), f p,γ =

[0,1]2

1 2p

<γ<

1 2

|f (x) − f (y)|2p dxdy |x − y|1+2pγ

1 − 2p , we deﬁne the

1 2p

.

We denote by Hp,γ the Banach space of continuous functions on [0, 1] vanishing at zero and having a ﬁnite (p, γ) norm. We deﬁne two families of random variables: W (s, ·) − W (s , ·)2p p,γ dsds Y 1 (σ) = |1+2pγ |s − s 2 [0,σ] and

Y 2 (τ ) = [0,τ ]2

W (·, t) − W (·, t )2p p,γ dtdt , |t − t |1+2pγ

where σ, τ ∈ [0, 1]. Set Y (σ, τ ) = Y 1 (σ) + Y 2 (τ ). We claim that there exists a constant R, depending on a, p, and γ, such that Y (σ, τ ) ≤ R

implies

sup

Wz ≤ a.

(2.36)

z∈[0,σ]×[0,1]∪[0,1]×[0,τ ]

In order to show this property, we ﬁrst apply Garsia, Rodemich, and Rumsey’s lemma (see Appendix, Lemma A.3.1) to the Hp,γ -valued function s → W (s, ·). From this lemma, and assuming Y 1 (σ) < R, we deduce 2pγ−1 W (s, ·) − W (s , ·)2p p,γ ≤ cp,γ R|s − s |

2.1 Regularity of densities and related topics

113

for all s, s ∈ [0, σ]. Hence, W (s, ·)2p p,γ ≤ cp,γ R for all s ∈ [0, σ]. Applying the same lemma to the real-valued function t → W (s, t) (s is now ﬁxed), we obtain |W (s, t) − W (s, t )|2p ≤ c2p,γ R|t − t |2pγ−1 for all t, t ∈ [0, 1]. Hence, 1

sup 0≤s≤σ,0≤t≤1

2p |W (s, t)| ≤ c1/p p,γ R .

Similarly, we can prove that 1

sup 0≤s≤1,0≤t≤τ

2p |W (s, t)| ≤ c1/p p,γ R ,

1/p

1

and it suﬃces to choose R in such a way that cp,γ R 2p < a. Now we introduce the stochastic process uA (s, t) and the random variable γ A that will verify the conditions of Deﬁnition 2.1.2. Let ψ : R+ → R+ be an inﬁnitely diﬀerentiable function such that ψ(x) = 0 if x > R, ψ(x) = 1 if x < R2 , and 0 ≤ ψ(x) ≤ 1. Then we deﬁne uA (s, t) = ψ(Y (s, t)) and

γA =

ψ(Y (s, t))dsdt. [0,1]2

On the set {M > a} we have (1) ψ(Y (s, t)) = 0 if (s, t) ∈ [0, Sa ] × [0, Ta ]. Indeed, if ψ(Y (s, t)) = 0, then Y (s, t) ≤ R (by deﬁnition of ψ) and by (2.36) this would imply supz∈[0,s]×[0,1]∪[0,1]×[0,t] Wz ≤ a, and, hence, s ≤ Sa , t ≤ Ta , which is contradictory. (2) Ds,t M = 1 if (s, t) ∈ [0, Sa ] × [0, Ta ], as we have proven before. Consequently, on {M > a} we obtain DM, uA H = Ds,t M ψ(Y (s, t))dsdt [0,1]2 = ψ(Y (s, t))dsdt = γ A . [0,Sa ]×[0,T Ta ]

114

2. Regularity of probability laws

We have γ A ∈ D∞ and uA ∈ D∞ (H) because the variables Y 1 (s) and Y 2 (t) are in D∞ (see Exercise 1.5.4 and [3]). So it remains to prove that γ −1 A has moments of all orders. We have ψ(Y (s, t))dsdt ≥ 1{Y (s,t)< R } dsdt [0,1]2

[0,1]2

2

= λ2 {(s, t) ∈ [0, 1]2 : Y 1 (s) + Y 2 (t) <

R } 2

R } 4 R ×λ1 {t ∈ [0, 1] : Y 2 (t) < } 4 1 −1 R 2 −1 R = (Y ) ( )(Y ) ( ). 4 4 ≥ λ1 {s ∈ [0, 1] : Y 1 (s) <

Here we have used the fact that the stochastic processes Y 1 and Y 2 are continuous and increasing. Finally for any we can write P ((Y 1 )−1 (

R ) < ) 4

R = P ( < Y 1 ()) 4 W (s, ·) − W (s , ·)2p R p,γ ≤ P dsds > |s − s |1+2pγ 4 [0,]2 p W (s, ·) − W (s , ·)2p 4 p p,γ ≤ ( ) E dsds 1+2pγ [0,]2 R |s − s | ≤ C2p

for some constant C > 0. This completes the proof of the theorem.

Exercises 2.1.1 Show that if F is a random variable in D2,4 such that E(DF −8 ) < DF ∞, then DF

2 ∈ Dom δ and δ

DF DF 2H

=−

DF ⊗ DF, D2 F H⊗H LF − 2 . DF 2H DF 4H

Hint: Show ﬁrst that DFDF belongs to Dom δ for any > 0 using

2H + Proposition 1.3.3, and then let tend to zero. 2.1.2 Let u = {ut , t ∈ [0, 1]} be an adapted continuous process belonging to L1,2 and such that sups,t∈[0,1] E[|Ds ut |2 ] < ∞. Show that if u1 = 0 a.s., 1 then the random variable F = 0 us dW Ws has an absolutely continuous law. 2.1.3 Suppose that F is a random variable in D1,2 , and let h be an element h belongs to the domain of δ. of H such that DF, hH = 0 a.s. and DF,h H

2.1 Regularity of densities and related topics

115

Show that F possesses a continuous and bounded density given by h . f (x) = E 1{F >x} δ DF, hH DF 2.1.4 Let F be a random variable in D1,2 such that Gk DF belongs to

2H Dom δ for any k = 0, . . . , n, where G0 = 1 and DF Gk = δ Gk−1 DF 2H

if 1 ≤ k ≤ n + 1. Show that F has a density of class C n and " ! f (k) (x) = (−1)k E 1{F >x} Gk+1 , 0 ≤ k ≤ n. 2.1.5 Let F ≥ 0 be a random variable in D1,2 such that Show that the density f of F veriﬁes 1 DF f p ≤ δ q (E(F )) p DF 2H

DF

DF 2H

∈ Dom δ.

for any p > 1, where q is the conjugate of p. 2.1.6 Let W = {W Wt , t ≥ 0} be a standard Brownian motion, and consider a random variable F in D1,2 . Show that for all t ≥ 0, except for a countable set of times, the random variable F + Wt has an absolutely continuous law (see [218]). 2.1.7 Let W = {W (s, t), (s, t) ∈ [0, 1]2 } be a two-parameter Wiener process. Show that for any pair of disjoint rectangles R1 , R2 with sides parallel to the axes we have P { sup W (z) = sup W (z)} = 0. z∈R1

z∈R2

Hint: Fix a rectangle [a, b] ⊂ [0, 1]2 . Show that the law of the random variable supz∈[a,b] W (z) conditioned by the σ-ﬁeld generated by the family {W (s, t), s ≤ a1 } is absolutely continuous. 2.1.8 Let F ∈ D3,α , α > 4, be a random variable such that E(DF −p H )< ∞ for all p ≥ 2. Show that the density p(x) of F is continuously diﬀerentiable, and compute p (x). 2.1.9 Let F = (F 1 , . . . , F m ) be a random vector whose components belong to the space D∞ . We denote by γ F the Malliavin matrix of F . Suppose that det γ F > 0 a.s. Show that the density of F is lower semicontinuous.

116

2. Regularity of probability laws

Hint: The density of F is the nondecreasing limit as N tends to inﬁnity of the densities of the measures [ΨN (γ F ) · P ] ◦ F −1 introduced in the proof of Theorem 2.1.1. 2

2.1.10 Let F = (W (h1 ) + W (h2 ))e−W (h2 ) , where h1 , h2 are orthonormal elements of H. Show that F ∈ D∞ , DF H > 0 a.s., and the density of F has a lower semicontinuous version satisfying p(0) = +∞ (see [197]). 1 Wt )dt, where W 2.1.11 Show that the random variable F = 0 t2 arctan(W is a Brownian motion, has a C ∞ density. 2.1.212 Let W = {W (s, t), (s, t) ∈ [0, 1]2 } be a two-parameter Wiener process. Show that the density of sup(s,t)∈[0,1]2 W (s, t) is strictly positive in (0, +∞). Hint: Apply Proposition 2.1.8.

2.2 Stochastic diﬀerential equations In this section we discuss the existence, uniqueness, and smoothness of solutions to stochastic diﬀerential equations. Suppose that (Ω, F, P ) is the canonical probability space associated with a d-dimensional Brownian motion {W i (t), t ∈ [0, T ], 1 ≤ i ≤ d} on a ﬁnite interval [0, T ]. This means Ω = C0 ([0, T ]; Rd ), P is the d-dimensional Wiener measure, and F is the completion of the Borel σ-ﬁeld of Ω with respect to P . The underlying Hilbert space here is H = L2 ([0, T ]; Rd ). Let Aj , B : [0, T ] × Rm → Rm , 1 ≤ j ≤ d, be measurable functions satisfying the following globally Lipschitz and boundedness conditions:

d for any (h1) j=1 |Aj (t, x) − Aj (t, y)| + |B(t, x) − B(t, y)| ≤ K|x − y|, x, y ∈ Rm , t ∈ [0, T ]; (h2) t → Aj (t, 0) and t → B(t, 0) are bounded on [0, T ]. We denote by X = {X(t), t ∈ [0, T ]} the solution of the following mdimensional stochastic diﬀerential equation: t d t X(t) = x0 + Aj (s, X(s))dW Wsj + B(s, X(s))ds, (2.37) j=1

0

0

where x0 ∈ Rm is the initial value of the process X. We will show that there is a unique continuous solution to this equation, such that for all t ∈ [0, T ] and for all i = 1, . . . , m the random variable X i (t) belongs to the space D1,p for all p ≥ 2. Furthermore, if the coeﬃcients are inﬁnitely diﬀerentiable in the space variable and their partial derivatives of all orders are uniformly bounded, then X i (t) belongs to D∞ . From now on we will use the convention of summation over repeated indices.

2.2 Stochastic diﬀerential equations

117

2.2.1 Existence and uniqueness of solutions Here we will establish an existence and uniqueness result for equations that are generalizations of (2.37). This more general type of equation will be satisﬁed by the iterated derivatives of the process X. Let V = {V (t), 0 ≤ t ≤ T } be a continuous and adapted M -dimensional stochastic process such that β p = sup E(|V (t)|p ) < ∞ 0≤t≤T

for all p ≥ 2. Suppose that σ : RM × Rm → Rm ⊗ Rd

b : RM × Rm → Rm

and

are measurable functions satisfying the following conditions, for a positive constant K: (h3) |σ(x, y) − σ(x, y )| + |b(x, y) − b(x, y )| ≤ K|y − y |, y, y ∈ Rm ;

for any x ∈ RM ,

(h4) the functions x → σ(x, 0) and x → b(x, 0) have at most polynomial growth order (i.e., |σ(x, 0)| + |b(x, 0)| ≤ K(1 + |x|ν ) for some integer ν ≥ 0). With these assumptions, we have the next result. Lemma 2.2.1 Consider a continuous and adapted m-dimensional process α = {α(t), 0 ≤ t ≤ T } such that dp = E(sup0≤t≤T |α(t)|p ) < ∞ for all p ≥ 2. Then there exists a unique continuous and adapted m-dimensional process Y = {Y (t), 0 ≤ t ≤ T } satisfying the stochastic diﬀerential equation

t

t

σ j (V (s), Y (s))dW Wsj +

Y (t) = α(t) + 0

b(V (s), Y (s))ds.

(2.38)

0

Moreover, E

sup |Y (t)|p

0≤t≤T

≤ C1

for any p ≥ 2, where C1 is a positive constant depending on p, T, K, β pν , m, and dp . Proof: Using Picard’s iteration scheme, we introduce the processes Y0 (t) = α(t) and

t

σ j (V (s), Yn (s))dW Wsj +

Yn+1 (t) = α(t) + 0

t

b(V (s), Yn (s))ds 0

(2.39)

118

2. Regularity of probability laws

if n ≥ 0. By a recursive argument one can show that Yn is a continuous and adapted process such that Yn (t)|p < ∞ (2.40) E sup |Y 0≤t≤T

for any p ≥ 2. Indeed, applying Doob’s maximal inequality (A.2) and Burkholder’s inequality (A.4) for m-dimensional martingales, and making use of hypotheses (h3) and (h4), we obtain Yn+1 (t)|p E sup |Y 0≤t≤T

p T ≤ cp dp + E σ j (V (s), Yn (s))dW Wsj 0 p ) T |b(V (s), Yn (s))| ds +E (

(

0

≤ cp dp + cp K p T p−1

)

T

(1 + E(|V (s)|νp ) + E(|Y Yn (s)|p )) ds 0

' & Yn (t)|p ) , ≤ cp dp + cp K p T p 1 + β νp + sup E(|Y 0≤t≤T

where cp and cp are constants depending only on p. Thus, Eq. (2.40) holds. Again applying Doob’s maximal inequality, Burkholder’s inequality, and condition (h3), we obtain, for any p ≥ 2,

E

Yn+1 (t) − Yn (t)| sup |Y

p

0≤t≤T

T

p

E (|Y Yn (s) − Yn−1 (s)| ) ds.

≤ cp K p T p−1 0

It follows inductively that the preceding expression is bounded by 1 (cp K p T p−1 )n+1 sup E(|Y Y1 (s)|p ). n! 0≤s≤T Consequently, we have ∞ n=0

E

p

sup |Y Yn+1 (t) − Yn (t)|

0≤t≤T

< ∞,

which implies the existence of a continuous process Y satisfying (2.38) and such that E(sup0≤t≤T |Y (t)|p ) ≤ C1 for all p ≥ 2. The uniqueness of the solution is derived by means of a similar method. As a consequence, taking V (t) = t in the Lemma 2.2.1 produces the following result.

2.2 Stochastic diﬀerential equations

119

Corollary 2.2.1 Assume that the coeﬃcients Aj and B of Eq. (2.37) are globally Lipschitz and have linear growth (conditions (h1) and (h2)). Then there exists a unique continuous solution X = {X(t), t ∈ [0, T ]} to Eq. (2.37). Moreover, sup |X(t)|p

E

0≤t≤T

≤ C1

for any p ≥ 2, where C1 is a positive constant depending on p, T, K, ν, and x0 .

2.2.2 Weak diﬀerentiability of the solution We will ﬁrst consider the case where the coeﬃcients Aj and B of the stochastic diﬀerential equation (2.37) are globally Lipschitz functions and have linear growth. Our aim is to show that the coordinates of the solution at each time t ∈ [0, T ] belong to the space D1,∞ = ∩p≥1 D1,p . To show this result we will make use of an extension of the chain rule to Lipschitz functions established in Proposition 1.2.4. We denote by Dtj (F ), t ∈ [0, T ], j = 1, . . . , d, the derivative of a random variable F as an element of L2 ([0, T ] × Ω; Rd ) L2 (Ω; H). Similarly we ,...,jN (F ) the N th derivative of F . denote by Dtj11,...,t N Using Proposition 1.2.4, we can show the following result. Theorem 2.2.1 Let X = {X(t), t ∈ [0, T ]} be the solution to Eq. (2.37), where the coeﬃcients are supposed to be globally Lipschitz functions with linear growth (hypotheses (h1) and (h2)). Then X i (t) belongs to D1,∞ for any t ∈ [0, T ] and i = 1, . . . , m. Moreover, sup E sup |Drj X i (s)|p < ∞, 0≤r≤t

and the derivative Drj X(t)

r≤s≤T

Drj X i (t)

satisﬁes the following linear equation: t = Aj (r, X(r)) + Ak,α (s)Drj (X k (s))dW Wsα r t + B k (s)Drj X k (s)ds

(2.41)

r

for r ≤ t a.e., and

Drj X(t) = 0

for r > t a.e., where Ak,α (s) and B k (s) are uniformly bounded and adapted m-dimensional processes. Proof:

Consider the Picard approximations given by X0 (t) = x0 ,

t

Aj (s, Xn (s))dW Wsj +

Xn+1 (t) = x0 + 0

t

B(s, Xn (s))ds (2.42) 0

120

2. Regularity of probability laws

if n ≥ 0. We will prove the following property by induction on n: (P) Xni (t) ∈ D1,∞ for all i = 1, . . . , m, n ≥ 0, and t ∈ [0, T ]; furthermore, for all p > 1 we have sup |Dr Xn (s)|p

ψ n (t) := sup E 0≤r≤t

<∞

(2.43)

s∈[r,t]

and

ψ n+1 (t) ≤ c1 + c2

t

ψ n (s)ds,

(2.44)

0

for some constants c1 , c2 . Clearly, (P) holds for n = 0. Suppose it is true for n. Applying Proposition 1.2.4 to the random vector Xn (s) and to the functions Aij and B i , we deduce that the random variables Aij (s, Xn (s)) and B i (s, Xn (s)) belong n,i

to D1,2 and that there exist m-dimensional adapted processes Aj (s) = n,i

n,i

(Aj,1 (s), . . . , Aj,m (s)) and B ded by K, such that

n,i

n,i

n,i

(s) = (B 1 (s), . . . , B m (s)), uniformly bounn,i

Dr [Aij (s, Xn (s))]

= Aj,k (s)Dr (Xnk (s))1{r≤s}

Dr [B i (s, Xn (s))]

= B k (s)Dr (Xnk (s))1{r≤s} .

(2.45)

and n,i

(2.46)

In fact, these processes are obtained as the weak limit of the sequences {∂k [Aij ∗ αm ](s, Xn (s)), m ≥ 1} and {∂k [B i ∗ αm ](s, Xn (s)), m ≥ 1}, where αm denotes an approximation of the identity, and it is easy to check the adaptability of the limit. From Proposition 1.5.5 we deduce that the random variables Aij (s, Xn (s)) and B i (s, Xn (s)) belong to D1,∞ . Thus the processes {Drl [Aij (s, Xn (s))], s ≥ r} and {Drl [B i (s, Xn (s))], s ≥ r} are square integrable and adapted, and from (2.45) and (2.46) we get |Dr [Aij (s, Xn (s))]| ≤ K|Dr Xn (s)|,

|Dr [B i (s, Xn (s))]| ≤ K|Dr Xn (s)|. (2.47) t Wsj Using Lemma 1.3.4 we deduce that the Itˆˆo integral 0 Aij (s, Xn (s))dW 1,2 belongs to the space D , and for r ≤ t we have t t Drl [ Aij (s, Xn (s))dW Wsj ] = Ail (r, Xn (r)) + Drl [Aij (s, Xn (s))]dW Wsj . 0

r

(2.48) On the other hand, 0 B i (s, Xn (s))ds ∈ D1,2 , and for r ≤ t we have t t Drl [ B i (s, Xn (s))ds] = Drl [B i (s, Xn (s))]ds. (2.49) t

0

r

2.2 Stochastic diﬀerential equations

121

i From these equalities and Eq. (2.42) we see that Xn+1 (t) ∈ D1,∞ for all t ∈ [0, T ], and we obtain

E

' & t j p−1 p p K E |Dr Xn (s)| ds , ≤ cp γ p + T

sup

r≤s≤t

|Drj Xn+1 (s)|p

r

(2.50) where γ p = sup E( sup |Aj (t, Xn (t))|p ) < ∞. n,j

0≤t≤T

So (2.43) and (2.44) hold for n + 1. From Lemma 2.2.1 we know that p E sup |Xn (s) − X(s)| −→ 0 s≤T

as n tends to inﬁnity. By Gronwall’s lemma applied to (2.50) we deduce that derivatives of the sequence Xni (t) are bounded in Lp (Ω; H) uniformly in n for all p ≥ 2. Therefore, from Proposition 1.5.5 we deduce that the random variables X i (t) belong to D1,∞ . Finally, applying the operator D to Eq. (2.37) and using Proposition 1.2.4, we deduce the linear stochastic diﬀerential equation (2.41) for the derivative of X i (t). If the coeﬃcients of Eq. (2.37) are continuously diﬀerentiable, then we can write i Ak,l (s) = (∂k Ail )(s, X(s)) and

i

B k (s) = (∂k B i )(s, X(s)). In order to prove the existence of higher-order derivatives, we will need the following technical lemma. Consider adapted and continuous processes α = {α(r, t), t ∈ [r, T ]} and V = {V Vj (t), 0 ≤ t ≤ T, j = 0, . . . , d} such that α is m-dimensional and Vj is uniformly bounded and takes values on the set of matrices of order m × m. Suppose that the random variables αi (r, t) and Vjkl (t) belong to D1,∞ for any i, j, k, l, and satisfy the following estimates: < ∞, sup E sup |α(r, t)|p 0≤r≤T

sup E 0≤s≤T

sup 0≤s,r≤T

E

r≤t≤T

sup |Ds Vj (t)|

< ∞,

p

s≤t≤T

sup r∨s≤t≤T

for any p ≥ 2 and any j = 0, . . . , d.

|Ds α(r, t)|p

< ∞,

122

2. Regularity of probability laws

Lemma 2.2.2 Let Y = {Y (t), r ≤ t ≤ T } be the solution of the linear stochastic diﬀerential equation

t

Vj (s)Y

Y (t) = α(r, t) +

(s)dW Wsj

+

r

t

V0 (s)Y (s)ds.

(2.51)

r

Then {Y i (t)} belongs to D1,∞ for any i = 1, . . . , m, and the derivative Ds Y i (t) veriﬁes the following linear equation, for s ≤ t: Dsj Y (t)

= Dsj α(r, t) + Vj (s)Y (s)1{r≤s≤t} t + [Dsj Vl (u)Y (u) + Vl (u)Dsj Y (u)]dW Wul r t + [Dsj V0 (u)Y (u) + V0 (u)Dsj Y (u)]du.

(2.52)

r

Proof: The proof can be done using the same technique as the proof of Theorem 2.2.1, and so we will omit the details. The main idea is to observe that Eq. (2.51) is a particular case of (2.38) when the coeﬃcients σ j and b are linear. Consider the Picard approximations deﬁned by the recursive equations (2.39). Then we can show by induction that the variables Yni (t) belong to D1,∞ and satisfy the equation Dsj Yn+1 (t)

=

Dsj α(r, t) + Vj (s)Y Yn (s)1{r≤s≤t} t + [Dsj Vl (u)Y Yn (u) + Vl (u)Dsj Yn (u)]dW Wul r t + [Dsj V0 (u)Y Yn (u) + V0 (u)Dsj Yn (u)]du. r

Finally, we conclude our proof as we did in the proof of Theorem 2.2.1. Note that under the assumptions of Lemma 2.2.2 the solution Y of Eq. (2.51) satisﬁes the estimates < ∞, E sup |Y (t)|p sup E 0≤s≤t

0≤t≤T

sup |Ds Y (t)|

p

r≤t≤T

< ∞,

for all p ≥ 2. Theorem 2.2.2 Let X be the solution of the stochastic diﬀerential equation (2.37), and suppose that the coeﬃcients Aij and B i are inﬁnitely differentiable functions in x with bounded derivatives of all orders greater than or equal to one and that the functions Aij (t, 0) and B i (t, 0) are bounded. Then X i (t) belongs to D∞ for all t ∈ [0, T ], and i = 1, . . . , m.

2.2 Stochastic diﬀerential equations

123

Proof: We know from Theorem 2.2.1 that for any i = 1, . . . , m and any t ∈ [0, T ], the random variable X i (t) belongs to D1,p for all p ≥ 2. Furthermore, the derivative Drj X i (t) veriﬁes the following linear stochastic diﬀerential equation: t j i i (∂k Ail )(s, X(s))Drj X k (s)dW Wsl Dr X (t) = Aj (r, Xr ) + r t + (∂k B)(s, X(s))Drj X k (s)ds. (2.53) r

Now we will recursively apply Lemma 2.2.2 to this linear equation. We N will denote by Drj11 ,...,j ,...,rN (X(t)) the iterated derivative of order N . We have to introduce some notation. For any subset K = {1 < · · · < η } of {1, . . . , N }, we put j(K) = j1 , . . . , jη and r(K) = r1 , . . . , rη . Deﬁne αil,j1 ,...,jN (s, r1 , . . . , rN ) = (∂k1 · · · ∂kν Ail )(s, X(s)) j(I )

j(I )

×Dr(I11 ) [X k1 (s)] · · · Dr(Iνν ) [X kν (s)] and

β ij1 ,...,jN (s, r1 , . . . , rN ) =

(∂k1 · · · ∂kν B i )(s, X(s)) j(I )

j(I )

×Dr(I11 ) [X k1 (s)] · · · Dr(Iνν ) [X kν (s)], where the sums are extended to the set of all partitions {1, . . . , N } = I1 ∪ · · · ∪ Iν . We also set αij (s) = Aij (s, X(s)). With these notations we will recursively show the following properties for any integer N ≥ 1: (P1) and

For any t ∈ [0, T ], p ≥ 2, and i = 1, . . . , m, X i (t) belongs to DN,p , p E sup |Dr1 ,...,rN (X(t))| < ∞. sup r1 ∨···∨rN ≤t≤T

r1 ,...,rN ∈[0,T ]

(P2)

The N th derivative satisﬁes the following linear equation:

i N Drj11 ,...,j ,...,rN (X (t)) =

N

αij ,j1 ,...,j−1 ,j+1 ,...,jN (r , r1 , . . . , r−1 , r+1 , . . . , rN )

=1

t

αil,j1 ,...,jN (s, r1 , . . . , rN )dW Wsl +β ij1 ,...,jN (s, r1 , . . . , rN )ds +

r1 ∨···∨rN

(2.54)

N if t ≥ r1 ∨ · · · ∨ rN , and Drj11 ,...,j ,...,rN (X(t)) = 0 if t < r1 ∨ · · · ∨ rN . We know that these properties hold for N = 1 because of Theorem 2.2.1. Suppose that the above properties hold up to the index N . Observe that αil,j1 ,...,jN (s, r1 , . . . , rN ) is equal to

k N (∂k Ail )(s, X(s))Drj11 ,...,j ,...,rN (X (s))

124

2. Regularity of probability laws

(this term corresponds to ν = 1) plus a polynomial function on the derivj(I) atives (∂k1 · · · ∂kν Ail )(s, X(s)) with ν ≥ 2, and the processes Dr(I) (X k (s)), with card(I) ≤ N − 1. Therefore, we can apply Lemma 2.2.2 to r = r1 ∨ · · · ∨ rN , and the processes Y (t) Vjik (t)

=

N Drj11 ,...,j ,...,rN (X(t)),

=

(∂k Aij )(s, X(s)),

t ≥ r, 1 ≤ i, k ≤ m,

j = 1, . . . , d,

and α(r, t) is equal to the sum of the remaining terms in the right-hand side of Eq. (2.54). Notice that with the above notations we have ! " Drj αil,j1 ,...,jN (t, r1 , . . . , rN ) = αil,j1 ,...,jN ,j (t, r1 , . . . , rN , r) and

! " Drj β ij1 ,...,jN (t, r1 , . . . , rN ) = β ij1 ,...,jN ,j (t, r1 , . . . , rN , r).

Using these relations and computing the derivative of (2.54) by means of Lemma 2.2.2, we obtain i N Drj Drj11 ,...,j ,...,rN (X (t))

=

N

αij ,j1 ,...,j−1 ,j+1 ,...,jN ,j (r , r1 , . . . , r−1 , r+1 , . . . , rN , r)

=1

+αij,j1 ,...,jN (r, r1 , . . . , rN ) t + Wsl αil,j1 ,...,jN ,j (s, r1 , . . . , rN , r)dW r1 ∨···∨rN +β ij1 ,...,jN ,j (s, r1 , . . . , rN , r)ds , which implies that property (P2) holds for N +1. The estimates of property (P1) are also easily derived. The proof of the theorem is now complete.

Exercises 2.2.1 Let σ and b be continuously diﬀerentiable functions on R with bounded derivatives. Consider the solution X = {Xt , t ∈ [0, T ]} of the stochastic diﬀerential equation t t Xt = x0 + σ(Xs )dW Ws + b(Xs )ds. 0

0

Show that for s ≤ t we have t t 1 Ds Xt = σ(Xs ) exp σ (Xs )dW Ws + [b − (σ )2 ](Xs )ds . 2 0 0

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

125

2.2.2 (Doss [84]) Suppose that σ is a function of class C 2 (R) with bounded ﬁrst and second partial derivatives and that b is Lipschitz continuous. Show that the one-dimensional stochastic diﬀerential equation t t 1 σ(Xs )dW Ws + [b + σσ ](Xs )ds (2.55) Xt = x0 + 2 0 0 has a solution that can be written in the form Xt = u(W Wt , Yt ), where (i) u(x, y) is the solution of the ordinary diﬀerential equation ∂u = σ(u), ∂x

u(0, y) = y;

(ii) for each ω ∈ Ω, {Y Yt (ω), t ≥ 0} is the solution of the ordinary diﬀerential equation Yt (ω) = f (W Wt (ω), Yt (ω)), Y0 (ω) = x0 , −1 x = b(u(x, y)) exp(− 0 σ (u(z, y)dz). where f (x, y) = b(u(x, y)) ∂u ∂y Using the above representation of the solution to Eq. (2.55), show that Xt belongs to D1,p for all p ≥ 2 and compute the derivative Ds Xt .

2.3 Hypoellipticity and H¨o¨rmander’s theorem In this section we introduce nondegeneracy conditions on the coeﬃcients of Eq. (2.37) and show that under these conditions the solution X(t) at any time t ∈ (0, T ] has a (smooth) density. Clearly, if the subspace spanned by {Aj (t, y), B(t, y); 1 ≤ j ≤ d, t ∈ [0, T ], y ∈ Rm } has dimension strictly smaller than m, then the law of X(t), for all t ≥ 0, will be singular with respect to the Lebesgue measure. We thus need some kind of nondegeneracy assumption.

2.3.1 Absolute continuity in the case of Lipschitz coeﬃcients Let {X(t), t ∈ [0, T ]} be the solution of the stochastic diﬀerential equation (2.37), where the coeﬃcients are supposed to be globally Lipschitz functions with linear growth. In Theorem 2.2.1 we proved that X i (t) belongs to D1,∞ for all i = 1, . . . , m and t ∈ [0, T ], and we found that the derivative Drj Xti satisﬁes the following linear stochastic diﬀerential equation: t i Ak,l (s)Drj Xsk dW Wsl Drj Xti = Aij (r, Xr ) + r t i + B k (s)Drj Xsk ds. (2.56) r

126

2. Regularity of probability laws

We are going to deduce a simpler expression for the derivative DXti . Consider the m × m matrix-valued process deﬁned by Yji (t) = δ ij +

t i i Ak,l (s)Y Yjk (s)dW Wsl + B k (s)Y Yjk (s)ds ,

(2.57)

0

i, j = 1, . . . , m. If the coeﬃcients of Eq. (2.37) are of class C 1+α , α > 0 (see Kunita [173]), then there is a version of the solution X(t, x0 ) to this equation that is continuously diﬀerentiable in x0 , and Y (t) is the Jacobian ∂X matrix ∂x (t, x0 ). 0 Now consider the m × m matrix-valued process Z(t) solution to the system t k Zki (s)Aj,l (s)dW Wsl Zji (t) = δ ij − 0 t k k α − Zki (s) B j (s) − Aα,l (s)Aj,l (s) ds.

(2.58)

0

By means of Itˆo’s formula, one can check that Zt Yt = Yt Zt = I. In fact, Zji (t)Y Ykj (t)

δ ik

t

j

Zji (s)Al,θ (s)Y Ykl (s)dW Wsθ 0 t t j l + Zji (s)B l (s)Y Ykl (s)ds − Zli (s)Aj,θ (s)Y Ykj (s)dW Wsθ 0 0 t l l α − Zli (s) B j (s) − Aα,θ (s)Aj,θ (s) Ykj (s)ds 0 t l j Zli (s)Aj,θ (s)Aα,θ (s)Y Ykα (s)ds = δ ik , − =

+

0

and similarly for Yt Zt . As a consequence, for any t ≥ 0 the matrix Yt is invertible and Yt−1 = Zt . Then it holds that Drj Xti = Yli (t)Y −1 (r)lk Akj (r, Xr ).

(2.59)

Indeed, it is enough to verify that the process {Y Yli (t)Y −1 (r)lk Akj (r, Xr ), t ≥ r} satisﬁes Eq. (2.56): Aij (r, Xr )

t

+

4 5 i β l Ak,l (s) Yαk (s)Y −1 (r)α β Aj (r, Xr ) dW (s)

r

4 5 i β B k (s) Yαk (s)Y −1 (r)α β Aj (r, Xr ) ds r ! " = Aij (r, Xr ) + Yli (t) − Yli (r) Y −1 (r)lθ Aθj (r, Xr ) t

+

=

Yli (t)Y −1 (r)lk Akj (r, Xr ).

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

127

We will denote by Qij t

=

DXti , DXtj H

=

d l=1

t

Drl Xti Drl Xtj dr

0

the Malliavin matrix of the vector X(t). Equation (2.59) allows us to write the following expression for this matrix: Qt = Yt Ct YtT ,

(2.60)

where Ctij =

d l=1

0

t

Y −1 (s)ik Akl (s, Xs )Y −1 (s)jk Akl (s, Xs )ds.

(2.61)

Deﬁne both the time-dependent m × m diﬀusion matrix σ ij (t, x) =

d

Aik (t, x)Ajk (t, x)

k=1

and the stopping time S = inf{t > 0 : 0

t

1{det σ(s,Xs )=0} ds > 0} ∧ T.

The following absolute continuity result has been established by Bouleau and Hirsch in [46]. Theorem 2.3.1 Let {X(t), t ∈ [0, T ]} be the solution of the stochastic differential equation (2.37), where the coeﬃcients are globally Lipschitz functions and of at most linear growth. Then for any 0 < t ≤ T the law of X(t) conditioned by {t > S} is absolutely continuous with respect to the Lebesgue measure on Rm . Proof: Taking into account Theorem 2.2.1 and Corollary 2.1.2, it suﬃces to show that det Qt > 0 a.s. on the set {t > S}. In view of expression (2.60) it is suﬃcient to prove that det Ct > 0 a.s. on this set. Suppose t > S. Then there exists a set G ⊂ [0, t] of positive Lebesgue measure such that for any s ∈ G and v ∈ Rm we have v T σ(s, Xs )v ≥ λ(s)|v|2 , where λ(s) > 0. Taking v = (Y Ys−1 )T u and integrating over [0, t] ∩ G, we obtain t uT Y (s)−1 σ(s, Xs )(Y (s)−1 )T uds ≥ k|u|2 , uT Ct u = t

0

where k = 0 1G (s) |Yλ(s) (s)|2 ds. Consequently, if t > S, the matrix Ct is invertible and the result is proved.

128

2. Regularity of probability laws

2.3.2 Absolute continuity under H¨rmander’s ¨ conditions In this section we assume that the coeﬃcients of Eq. (2.37) are inﬁnitely diﬀerentiable with bounded derivatives of all orders and do not depend on the time. Let us denote by X = {X(t), t ≥ 0} the solution of this equation on [0, ∞). We have seen in Theorem 2.2.2 that in such a case the random variables X i (t) belong to the space D∞ . We are going to impose nondegeneracy conditions on the coeﬃcients in such a way that the solution has a smooth density. To introduce these conditions, consider the following vector ﬁelds on Rm associated with the coeﬃcients of Eq. (2.37): ∂ , ∂xi ∂ = B i (x) . ∂xi

= Aij (x)

Aj B

j = 1, . . . , d,

The covariant derivative of Ak in the direction of Aj is deﬁned as the vector l i ∂ ﬁeld A∇ j Ak = Aj ∂l Ak ∂xi , and the Lie bracket between the vector ﬁelds Aj and Ak is deﬁned by ∇ [Aj , Ak ] = A∇ j Ak − Ak Aj .

Set A0

=

& ' ∂ 1 B i (x) − Ajl (x)∂ ∂j Ail (x) 2 ∂xi 1 ∇ Al Al . 2 d

= B−

l=1

The vector ﬁeld A0 appears when we write the stochastic diﬀerential equation (2.37) in terms of the Stratonovich integral instead of the Itˆ o integral: t t Aj (Xs ) ◦ dW Wsj + A0 (Xs )ds. Xt = x0 + 0

0

H¨ ormander’s condition can be stated as follows: (H)

The vector space spanned by the vector ﬁelds

A1 , . . . , Ad ,

[Ai , Aj ], 0 ≤ i, j ≤ d,

[Ai , [Aj , Ak ]], 0 ≤ i, j, k ≤ d, . . .

at point x0 is Rm . For instance, if m = d = 1, A11 (x) = a(x), and A10 (x) = b(x), then H¨ ormander’s condition means that a(x0 ) = 0 or an (x0 )b(x0 ) = 0 for some n ≥ 1. In this situation we have the following result. Theorem 2.3.2 Assume that H¨ ormander’s condition (H) holds. Then for any t > 0 the random vector X(t) has a probability distribution that is absolutely continuous with respect to the Lebesgue measure.

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

129

We will see in the next section that the density of the law of Xt is inﬁnitely diﬀerentiable on Rm . This result can be considered as a probabilistic version of Hormander’s ¨ theorem on the hypoellipticity of second-order differential operators. Let us discuss this point with some detail. We recall that a diﬀerential operator A on an open set G of Rm with smooth (i.e., inﬁnitely diﬀerentiable) coeﬃcients is called hypoelliptic if, whenever u is a distribution on G, u is a smooth function on any open set G ⊂ G on which Au is smooth. Consider the second-order diﬀerential operator 1 (Ai )2 + A0 . 2 i=1 d

A=

(2.62)

Hormander’s ¨ theorem [138] states that if the Lie algebra generated by the vector ﬁelds A0 , A1 , . . . , Ad has full rank at each point of Rm , then the operator L is hypoelliptic. Notice that this assumtion is stronger than (H). A straightforward proof of this result using the calculus of pseudo-diﬀerential operators can be found in Khon [170]. On the other hand, Ole˘nik ˘ and Radkeviˇc [277] have made generalizations of H¨o¨rmander’s theorem to include operators L, which cannot be written in Hormander’s ¨ form (as a sum of squares). In order to relate the hypoellipticity property with the smoothness of the density of Xt , let us consider an inﬁnitely diﬀerentiable function f with compact support on (0, ∞) × Rm . By means of Itˆo’s formula we can write for t large enough & ' t ∂ + G)f (s, Xs )ds , 0 = E[f (t, Xt )] − E[f (0, X0 )] = E ( 0 ∂s where G=

m m ∂2 ∂ 1 (AAT )ij + Bi . 2 i,j=1 ∂xi ∂xj ∂xi i=1

Notice that G − B = L − A0 , where L is deﬁned in (2.62). Denote by pt (dy) the probability distribution of Xt . We have & ' ∞ ∞ ∂ ∂ + G)f (s, Xs )ds = + G)f (s, y)ps (dy)ds. ( ( 0=E ∂s 0 0 Rm ∂s ∂ + This means that pt (dy) satisﬁes the forward Fokker-Planck equation (− ∂t ∗ ∗ G )p = 0 (where G denotes the adjoint of the operator G) in the distribution sense. Therefore, the fact that pt (dy) has a C ∞ density in the variable ∂ −G ∗ . Increasing y is implied by the hypoelliptic character of the operator ∂t the dimension by one and applying H¨ o¨rmander’s theorem to the operator

130

2. Regularity of probability laws

− G ∗ , one can deduce its hypoellipticity assuming hypothesis (H) at each point x0 in Rm . We refer to Williams [350] for a more detailed discussion of this subject. ∂ ∂t

Let us turn to the proof of Theorem 2.3.2. First we carry out some preliminary computations that will explain the role played by the nonde∂ is a C ∞ vector generacy condition (H). Suppose that V (x) = V i (x) ∂x i m o’s formula to the ﬁeld on R . The Lie brackets appear when we apply Itˆ process Yt−1 V (Xt ), where the process Yt−1 has been deﬁned in (2.58). In fact, we have t Yt−1 V (Xt ) =V (x0 ) + Ys−1 [Ak , V ](Xs )dW Wsk 0 3 2 t d 1 −1 + Ys [Ak , [Ak , V ]] (Xs )ds. (2.63) [A0 , V ] + 2 0 k=1

We recall that from (2.58) we have Yt−1

= I−

d k=1 t

−

t

0

Ys−1 ∂Ak (Xs )dW Wsk

(

Ys−1

∂B(Xs ) −

0

d

) ∂Ak (Xs )∂Ak (Xs ) ds,

k=1

i wherei ∂Ak and ∂B respectively denote the Jacobian matrices ∂j Ak and ∂j B , i, j = 1, . . . , m. In order to show Eq. (2.63), we ﬁrst use Itˆ oˆ’s formula: t d Yt−1 V (Xt ) = V (x0 ) + Ys−1 (∂V Ak − ∂Ak V ) (Xs )dW Wsk 0

k=1 t

Ys−1 (∂V B − ∂BV ) (Xs )ds

+ 0

t

Ys−1

+ 0

+

1 2

−

t

Ys−1

0 t

(∂Ak ∂Ak V )(Xs )ds

(2.64)

k=1

d

Ys−1

0

m

∂i ∂j V (Xs )

i,j=1 d

d

Aik (Xs )Ajk (Xs )ds

k=1

(∂Ak ∂V Ak )(Xs )ds.

k=1

Notice that ∂V Ak − ∂Ak V

=

[Ak , V ],

∂V B − ∂BV

=

[B, V ].

and

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

131

Additionally, we can write 1 [Ak , [Ak , V ]] − [B, V ] 2 k=1 ( ) d d 1 ∇ 1 = − Ak Ak , V + [Ak , [Ak , V ]] 2 2 d

[A0 , V ] +

k=1

=

1 2

d

k=1

9 ∇ ∇ ∇ ∇ ∇ −(A∇ k Ak ) V + V (Ak Ak ) + Ak (Ak V )

k=1 ∇ −A∇ k (V Ak )

∇ ∇ ∇ − (A∇ k V ) Ak + (V Ak ) Ak

:

1 4 − Aik ∂i Alk ∂l V + V i ∂i Alk ∂l Ak + V i Alk ∂i ∂l Ak 2 d

=

k=1

+Aik ∂i Alk ∂l V + Aik Alk ∂i ∂l V − Aik ∂i V l ∂l Ak −Aik V l ∂i ∂l Ak − Aik ∂i V l ∂l Ak + V i ∂i Alk ∂l Ak =

5

d 4 5 1 V i ∂i Alk ∂l Ak + Aik Alk ∂i ∂l V − Aik ∂i V l ∂l Ak . 2

k=1

Finally expression (2.63) follows easily from the previous computations. Proof of Theorem 2.3.2: Fix t > 0. Using Theorem 2.1.2 (or Theorem 2.1.1) it suﬃces to show that the matrix Ct given by (2.61) is invertible with probability one. Suppose that P {det Ct = 0} > 0. We want to show that under this assumption condition (H) cannot be satisﬁed. Let Ks be Yσ−1 Ak (Xσ ); 0 ≤ σ ≤ s, k = the random subspace of Rm spanned by {Y 1, . . . , d}. The family of vector spaces {Ks , s ≥ 0} is increasing. Set K0+ = ∩s>0 Ks . By the Blumenthal zero-one law for the Brownian motion (see Revuz and Yor [292, Theorem III.2.15]), K0+ is a deterministic space with probability one. Deﬁne the increasing adapted process {dim Ks , s > 0} and the stopping time τ = inf{s > 0 : dim Ks > dim K0+ }. Notice that P {τ > 0} = 1. For any vector v ∈ Rm of norm one we have v T Ct v =

d k=1

t

|v T Ys−1 Ak (Xs )|2 ds.

0

As a consequence, by continuity v T Ct v = 0 implies v T Ys−1 Ak (Xs ) = 0 for any s ∈ [0, t] and any k = 1, . . . , d. Therefore, K0+ = Rm , otherwise Ks = Rm for any s > 0 and any vector v verifying v T Ct v = 0 would be equal to zero, which implies that Ct is invertible a.s., in contradiction with

132

2. Regularity of probability laws

our hypothesis. Let v be a ﬁxed nonzero vector orthogonal to K0+ . Observe that v⊥Ks if s < τ , that is, v T Ys−1 Ak (Xs ) = 0,

for k = 1, . . . , d

and s < τ .

(2.65)

We introduce the following sets of vector ﬁelds: Σ0 Σn

= {A1 , . . . , Ad }, = {[Ak , V ], k = 1, . . . , d, V ∈ Σn−1 }

Σ =

if n ≥ 1,

∪∞ n=0 Σn ,

and Σ0 Σn

= Σ0 , = {[Ak , V ], k = 1, . . . , d, V ∈ Σn−1 ; 1 [A0 , V ] + [Aj , [Aj , V ]], V ∈ Σn−1 } if n ≥ 1, 2 j=1 d

Σ

= ∪∞ n=0 Σn .

We denote by Σn (x) (resp. Σn (x)) the subset of Rm obtained by freezing the variable x in the vector ﬁelds of Σn (resp. Σn ). Clearly, the vector spaces o¨rmander’s condition spanned by Σ(x) or by Σ (x) coincide, and under H¨ this vector space is Rm . We will show that for all n ≥ 0 the vector v is oro¨rmander’s condition. thogonal to Σn (x0 ), which is in contradiction with H¨ This claim will follow from the following stronger orthogonality property: v T Ys−1 V (Xs ) = 0,

for all s < τ , V ∈ Σn , n ≥ 0.

(2.66)

Indeed, for s = 0 we have Y0−1 V (X0 ) = V (x0 ). Property (2.66) can be proved by induction on n. For n = 0 it reduces to (2.65). Suppose that it holds for n − 1, and let V ∈ Σn−1 . Using formula (2.63) and the induction hypothesis, we obtain s v T Yu−1 [Ak , V ](Xu )dW Wuk 0 = 0 2 3 s d 1 T −1 + v Yu [A0 , V ] + [Ak , [Ak , V ]] (Xu )du 2 0 k=1

for s < τ . If a continuous semimartingale vanishes in a random interval [0, τ ), where τ is a stopping time, then the quadratic variation of the martingale part and the bounded variation part of the semimartingale must be zero on this interval. As a consequence we obtain v T Ys−1 [Ak , V ](Xs ) = 0

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

and

133

2 v

T

Ys−1

3 d 1 [A0 , V ] + [Ak , [Ak , V ]] (Xs ) = 0, 2 k=1

for any s < τ . Therefore (2.66) is true for n, and the proof of the theorem is complete.

2.3.3 Smoothness of the density under H¨ o¨rmander’s condition In this section we will show the following result. Theorem 2.3.3 Assume that {X(t), t ≥ 0} is the solution to Eq. (2.37), where the coeﬃcients do not depent on the time. Suppose that the coeﬃcients Aj , 1 ≤ j ≤ d, B are inﬁnitely diﬀerentiable with bounded partial derivatives of all orders and that Hormander’s H¨ condition (H) holds. Then for any t > 0 the random vector X(t) has an inﬁnitely diﬀerentiable density. From the previous results it suﬃces to show that (det Ct )−1 has moments of all orders. We need the following preliminary lemmas. Lemma 2.3.1 Let C be a symmetric nonnegative deﬁnite m × m random matrix. Assume that the entries C ij have moments of all orders and that for any p ≥ 2 there exists 0 (p) such that for all ≤ 0 (p) sup P {v T Cv ≤ } ≤ p .

|v|=1

Then (det Ct )−1 ∈ Lp (Ω) for all p. Proof: Let λ = inf |v|=1 v T Cv be the smallest eigenvalue of C. We know that λm ≤ det C. Thus, it suﬃces to show that E(λ−p ) < ∞ for all p ≥ 2. 1

m ij 2 2 (C ) . Fix > 0, and let v1 , . . . , vN be a ﬁnite set of Set |C| = i,j=1 unit vectors such that the balls with their center in these points and radius 2 m−1 . Then we have 2 cover the unit sphere S P {λ < }

= P { inf v T Cv < } |v|=1

≤ P { inf v T Cv < , |C| ≤ |v|=1

1 1 } + P {|C| > }. (2.67)

Assume that |C| ≤ 1 and vkT Cvk ≥ 2 for any k = 1, . . . , N . For any unit 2 vector v there exists a vk such that |v − vk | ≤ 2 and we can deduce the

134

2. Regularity of probability laws

following inequalities: ≥ vkT Cvk − |v T Cv − vkT Cvk | ! " ≥ 2 − |v T Cv − v T Cvk | + |v T Cvk − vkT Cvk | ≥ 2 − 2|C||v − vk | ≥ .

v T Cv

As a consequence, (2.67) is bounded by

N 8

{vkT Cvk k=1

P

< 2}

+ P {|C| >

1 } ≤ N (2)p+2m + p E(|C|p )

if ≤ 12 0 (p + 2m). The number N depends on but is bounded by a constant times −2m . Therefore, we obtain P {λ < } ≤ const.p for all ≤ 1 (p) and for all p ≥ 2. Clearly, this implies that λ−1 has moments of all orders. The next lemma has been proved by Norris in [239], following the ideas of Stroock [320], and is the basic ingredient in the proof of Theorem 2.3.3. The heuristic interpretation of this lemma is as follows: It is well known that if the quadratic variation and the bounded variation component of a continuous semimartingale vanish in some time interval, then the semimartingale vanishes in this interval. (Equation (2.69) provides a quantitative version of this result.) That is, when the quadratic variation or the bounded variation part of a continuous semimartingale is large, then the semimartingale is small with an exponentially small probability. Lemma 2.3.2 Let α, y ∈ R. Suppose that β(t), γ(t) = (γ 1 (t), . . . , γ d (t)), and u(t) = (u1 (t), . . . , ud (t)) are adapted processes. Set a(t)

=

t

0 t

Y (t)

=

t

γ i (s)dW Wsi

β(s)ds +

α+ y+

0 t

ui (s)dW Wsi ,

a(s)ds + 0

0

and assume that there exists t0 > 0 and p ≥ 2 such that

c=E

sup (|β(t)| + |γ(t)| + |a(t)| + |u(t)|)

0≤t≤t0

p

< ∞.

(2.68)

Then, for any q > 8 and for any r, ν > 0 such that 18r + 9ν < q − 8, there exists 0 = 0 (t0 , q, r, ν) such that for all ≤ 0 P 0

t0

Yt2 dt

t0

q

< , 0

(|a(t)| + |u(t)| )dt ≥ 2

2

≤ crp + e−

−ν

. (2.69)

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

135

Proof: Set θt = |β(t)| + |γ(t)| + |a(t)| + |u(t)|. Fix q > 8 and r, ν such that 18r + 9ν < q − 8. Suppose that ν < ν also satisﬁes 18r + 9ν < q − 8 Then we deﬁne the bounded stopping time −r T = inf s ≥ 0 : sup θu > ∧ t0 . 0≤u≤s

We have

P

t0

Yt2 dt

t0

q

< ,

0

(|a(t)| + |u(t)| )dt ≥ ≤ A1 + A2 , 2

2

0

with A1 = P {T < t0 } and t0 2 q Yt dt < , A2 = P 0

t0

(|a(t)| + |u(t)| )dt ≥ , T = t0 . 2

2

0

By the deﬁnition of T and condition (2.68), we obtain & ' p −r rp sup θs > A1 ≤ P ≤ E sup θs ≤ crp . 0≤s≤t0

0≤s≤t0

Let us introduce the following notation: t t At = a(s)ds, Mt = ui (s)dW Wsi , 0 0 t t Y (s)ui (s)dW Wsi , Qt = A(s)γ i (s)dW Wsi . Nt = 0

0

Deﬁne for any ρi > 0, δ i > 0, i = 1, 2, 3, Ns | ≥ δ 1 , B1 = N T < ρ1 , sup |N 0≤s≤T

B2 = M T < ρ2 , sup |M Ms | ≥ δ 2 , 0≤s≤T

B3 = QT < ρ3 , sup |Qs | ≥ δ 3 . 0≤s≤T

By the exponential martingale inequality (cf. (A.5)), P (Bi ) ≤ 2 exp(−

δ 2i ), 2ρi

(2.70)

for i = 1, 2, 3. Our aim is to prove the following inclusion: 2 3 T T 2 q 2 2 Yt dt < , (|a(t)| + |u(t)| )dt ≥ , T = t0 0

0

⊂ B1 ∪ B2 ∪ B3 ,

(2.71)

136

2. Regularity of probability laws

for the particular choices of ρi and δ i : ρ1 = −2r+q , 1

ρ2 = 2(2t0 + 1) 2 −2r+

q1 2

,

ρ3 = 36−2r+2q2 t0 ,

δ 1 = q1 ,

q1 =

q 2

−r−

δ 2 = q2 ,

q2 =

q 8

−

δ 3 = q3 ,

q3 =

q 8

−

5r 4 9r 4

ν 2 , 5ν 8 ,

− −

9ν 8 .

From the inequality (2.70) and the inclusion (2.71) we get δ 21 δ 22 δ 23 A2 ≤ 2 exp(− ) + exp(− ) + exp(− ) 2ρ1 2ρ2 2ρ3 1 −ν 1 1 −ν −ν ≤ 2 exp(− ) + exp(− √ ) + exp(− ) 2 72t0 4 1 + 2t0 ≤ exp(−−ν ) for ≤ 0 , because = −ν ,

2q1 + 2r − q q1 2q2 + 2r − 2 2q3 + 2r − 2q2

= −ν , = −ν ,

which allows us to complete the proof of the lemma. It remains only to check the inclusion (2.71). T Proof of (2.71): Suppose that ω ∈ B1 ∪ B2 ∪ B3 , T (ω) = t0 , and 0 Yt2 dt < q . Then T

N T = 0

Yt2 |ut |2 dt < −2r+q = ρ1 .

t Then since ω ∈ B1 , sup0≤s≤T 0 Ys uis dW Wsi < δ 1 = q1 . Also t Ys as ds ≤ t0 sup 0≤s≤T 0

Thus,

12

T

Yt2 a2t dt

0

1

q

< t02 −r+ 2 .

t √ q Ys dY Ys < t0 −r+ 2 + q1 . sup 0≤s≤T 0

By Itˆˆo’s formula

Yt2

2

=y +2

T

M t dt

=

0

0

Ys dY Ys + M t , and therefore

T

T

t

− Ty − 2 Ys dY Ys dt 0 0 √ q q + 2t0 t0 −r+ 2 + q1 < (2t0 + 1)q1 , Yt2 dt

0

<

t

2

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

for ≤ 0 because q > q1 and −r + process, for any 0 < γ < T we have

q 2

137

> q1 . Since M t is an increasing

γM T −γ < (2t0 + 1)q1 , 1

and hence M T < γ −1 (2t0 + 1)q1 + γ−2r . Choosing γ = (2t0 + 1) 2 we obtain M T < ρ2 , provided < 1. Since ω ∈ B2 we get

q1 2

,

sup |M Mt | < δ 2 = q2 .

0≤s≤T

Recall that

T 0

Yt2 dt < q so that, by Tchebychev’s inequality, q

q

Yt (ω)| ≥ 3 } ≤ 3 , λ1 {t ∈ [0, T ] : |Y and therefore q

q

λ1 {t ∈ [0, T ] : |y + At (ω)| ≥ 3 + q2 } ≤ 3 . q

We can assume that 3 < t20 , provided ≤ 0 (t0 ). So for each t ∈ [0, T ], q q there exists s ∈ [0, T ] such that |s − t| ≤ 3 and |y + As | < 3 + q2 . Consequently, t q ar dr| < (1 + −r ) 3 + q2 . |y + At | ≤ |y + As | + | s −r

q 3

) + , and for all t ∈ [0, T ] we have q |At | < 2 (1 + −r ) 3 + q2 ≤ 6q2 ,

In particular, |y| < (1 +

because q2 <

q 3

q2

− r. This implies that T QT = A2t |γ t |2 dt < 36t0 2q2 −2r = ρ3 . 0

So since ω ∈ B3 , we have

T |QT | = At γ i (t)dW Wi (t) < δ 3 = q3 . 0

Finally, by Itˆ oˆ’s formula we obtain T T (a2t + |ut |2 )dt = at dAt + M T 0

0

= aT AT −

T

T

At β t dt − 0

At γ i (t)dW Wti + M T 0

√ q1 ≤ (1 + t0 )6q2 −r + q3 + 2 2t0 + 1−2r+ 2 < for ≤ 0 , because q2 − r > q3 , q3 > 1, and −2r +

q1 2

> 1.

138

2. Regularity of probability laws

Now we can proceed to the proof of Theorem 2.3.3. Proof of Theorem 2.3.3: Fix t > 0. We want to show that E[(det Ct )−p ] < ∞ for all p ≥ 2. By Lemma 2.3.1 it suﬃces to see that for all p ≥ 2 we have sup P {v T Ct v ≤ } ≤ p

|v|=1

for any ≤ 0 (p). We recall the following expression for the quadratic form associated to the matrix Ct : d t T −1 v Ys Aj (Xs )2 ds. v Ct v = T

j=1

0

By Hormander’s ¨ condition, there exists ;j0 an integer j0 ≥ 0 such that the Σj (x) at point x0 has dimension linear span of the set of vector ﬁelds j=0 m. As a consequence there exist constants R > 0 and c > 0 such that j0

(v T V (y))2 ≥ c,

j=0 V ∈Σj

for all v and y with |v| = 1 and |y − x0 | < R. For any j = 0, 1, . . . , j0 we put m(j) = 2−4j and we deﬁne the set ⎧ ⎫ ⎨ t ⎬ 2 v T Ys−1 V (Xs ) ds ≤ m(j) . Ej = ⎩ ⎭ 0 V ∈Σj

Notice that {v T Ct v ≤ } = E0 because m(0) = 1. Consider the decomposition Ej0 −1 ∩ Ejc0 ) ∪ F, E0 ⊂ (E0 ∩ E1c ) ∪ (E1 ∩ E2c ) ∪ · · · ∪ (E where F = E0 ∩ E1 ∩ · · · ∩ Ej0 . Then for any unit vector v we have P {v Ct v ≤ } = P (E0 ) ≤ P (F ) + T

j0

P (E Ej ∩ Ejc+1 ).

j=0

We are going to estimate each term of this sum. This will be done in two steps. Step 1:

Consider the following stopping time:

S = inf{σ ≥ 0 : sup |Xs − x0 | ≥ R 0≤s≤σ

or

sup |Y Ys−1 − I| ≥

0≤s≤σ

We can write P (F ) ≤ P F ∩ {S ≥ β } + P {S < β },

1 } ∧ t. 2

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

139

where 0 < β < m(j0 ). For small enough, the intersection F ∩ {S ≥ β } is empty. In fact, if S ≥ β , we have j0 t T −1 2 v Ys V (Xs ) ds 0

j=0 V ∈Σj

≥

j0

S

0

j=0 V ∈Σj

v T Ys−1 V (Xs ) |v T Ys−1 |

2

|v T Ys−1 |2 ds ≥

cβ , (2.72) 4

because s < S implies |v T Ys−1 | ≥ 1 − |I − Ys−1 | ≥ 12 . On the other hand, the left-hand side of (2.72) is bounded by (j0 + 1)m(j0 ) on the set F , and for small enough we therefore obtain F ∩ {S ≥ β } = ∅. Moreover, it holds that 2 3 P {S < β }

≤ P

sup |Xs − x0 | ≥ R

0≤s≤β

2 +P

sup 0≤s≤β

|Y Ys−1

(

≤ R

−q

1 − I| ≥ 2 )

sup |Xs − x0 |

q

E

3 ( q

+2 E

0≤s≤β

) sup 0≤s≤β

|Y Ys−1

− I|

q

for any q ≥ 2. Now using Burkholder’s and H¨ older’s inequalities, we deduce qβ that P {S < β } ≤ C 2 for any q ≥ 2, which provides the desired estimate for P (F ). For any j = 0, . . . , j0 we introduce the following probability: ⎧ ⎨ t 2 c v T Ys−1 V (Xs ) ds ≤ m(j) , P (E Ej ∩ Ej +1 ) = P ⎩ V ∈Σj 0 ⎫ ⎬ t 2 v T Ys−1 V (Xs ) ds > m(j+1) ⎭ 0 Step 2:

V ∈Σj

≤

P

t

2 v T Ys−1 V (Xs ) ds ≤ m(j) ,

0

V ∈Σj

d t T −1 2 v Ys [Ak , V ](Xs ) ds +

k=1

+

0

d 1

2

j=1

⎞

⎞2

t 0

v T Ys−1 [A0 , V ] ⎫ ⎪

m(j+1) ⎬

[Aj , [Aj , V ]]⎠ (Xs )⎠ ds >

n(j) ⎪ ⎭

,

140

2. Regularity of probability laws

where n(j) denotes the cardinality of the set Σj . Consider the continuous semimartingale {v T Ys−1 V (Xs ), s ≥ 0}. From (2.63) we see that the quadratic variation of this semimartingale is equal to d k=1

s

2 v T Yσ−1 [Ak , V ](Xσ ) dσ,

0

and the bounded variation component is ⎫ ⎧ s d ⎬ ⎨ 1 v T Yσ−1 [A0 , V ] + [Aj , [Aj , V ]] (Xσ )dσ. ⎭ ⎩ 2 0 j=1

Taking into account that 8m(j + 1) < m(j), we get the desired estimate from Lemma 2.3.2 applied to the semimartingale Yt = v T Ys−1 V (Xs ). The proof of the theorem is now complete. Remarks:

d 1. Note that if the diﬀusion matrix σ(x) = j=1 Aj (x)ATj (x) is elliptic at ormander’s condition (H) holds, the initial point (that is, σ(x0 ) > 0), then H¨ and for any t > 0 the random variable Xt has an inﬁnitely diﬀerentiable density. The interesting applications of H¨ ormander’s theorem appear when σ(x0 ) is degenerate. Consider the following elementary example. Let m = d = 2, X0 = 0, B = 0, and consider the vector ﬁelds & & ' ' 1 sin x2 A1 (x) = and A2 (x) = . x1 2x1 In this case the diﬀusion matrix & ' 1 + sin2 x2 x1 (2 + sin x2 ) σ(x) = 5x21 x1 (2 + sin x2 ) degenerates along ' the line x1 = 0. The Lie bracket [A1 , A2 ] is equal to & 2x1 cos x2 . Therefore, the vector ﬁelds A1 and [A1 , A2 ] at x = 0 span 1 − 2 sin x2 ormander’s condition holds. So from Theorem 2.3.3 X(t) has a R2 and H¨ C ∞ density for any t > 0. 2. The following is a stronger version of H¨ ormander’s condition: (H1) The Lie algebra space spanned by the vector ﬁelds A1 , . . . , Ad at point x0 is Rm . The proof of Theorem 2.3.3 under this stronger hypothesis can be done using the simpler version of Lemma 2.3.2 stated in Exercise 2.3.4.

2.3 Hypoellipticity and H¨ ¨ ormander’s theorem

141

Exercises 2.3.1 Let W = {(W 1 , W 2 ), t ≥ 0} be a two-dimensional Brownian motion, and consider the process X = {Xt , t ≥ 0} deﬁned by Xt1 = Wt1 , t 2 Ws1 dW Ws2 . Xt = 0

Compute the Malliavin matrix γ t of the vector Xt , and show that t det γ t ≥ t (W Ws1 )2 ds. 0

t

Using Lemma 2.3.2 show that E[| 0 (W Ws1 )2 ds|−p ] < ∞ for all p ≥ 2, and conclude that for all t > 0 the random variable Xt has an inﬁnitely diﬀerentiable density. Obtain the same result by applying Theorem 2.3.3 to a stochastic diﬀerential equation satisﬁed by X(t). 2.3.2 Let f (s, t) be a square integrable symmetric kernel on [0, 1]. Set F = I2 (f ). Show that the norm of the derivative of F is given by DF 2H =

∞

λ2n W (en )2 ,

n=1

where {λn } and {en } are the corresponding sequence of eigenvalues and orthogonal eigenvectors of the operator associated with f . In the particular case where λn = (πn)−2 , show that P (DF H < ) ≤

√

2 exp(−

1 ), 82

and conclude that F has an inﬁnitely diﬀerentiable density. 2 2 Hint: Use Tchebychev’s exponential inequality with the function e−λ x and then optimize over λ. 2.3.3 Let m = 3, d = 2, and X0 = 0, and consider the vector ﬁelds ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ 0 1 0 A1 (x) = ⎣ 0 ⎦ , A2 (x) = ⎣ sin x2 ⎦ , B(x) = ⎣ 12 sin x2 cos x2 + 1 ⎦ . x1 0 1 Show that the solution to the stochastic diﬀerential equation X(t) associated to these coeﬃcients has a C ∞ density for any t > 0. 2.3.4 Prove the following stronger version of Lemma 2.3.2: Let t t Y (t) = y + a(s)ds + ui (s)dW Wsi , t ∈ [0, t0 ], 0

0

142

2. Regularity of probability laws

be a continuous semimartingale such that y ∈ R and a and ui are adapted processes verifying & ' c := E sup (|at | + |ut |)p < ∞. 0≤t≤t0

Then for any q, r, ν > 0 verifying q > ν + 10r + 1 there exists 0 = 0 (t0 , q, r, ν) such that for ≤ 0 t0 t0 −ν P Yt2 dt < q , |u(t)|2 dt ≥ ≤ crp + e− . 0

0

2.3.5 (Elworthy formula [90]) Let X = {X(t), t ∈ [0, T ]} be the solution to the following d-dimensional stochastic diﬀerential equation: X(t) = x0 +

d j=1

0

t

Aj (X(s))dW Wsj

+

t

B(X(s))ds, 0

where the coeﬃcients Aj and B are of class C 1+α , α > 0, with bounded derivatives. We also assume that the m × m matrix A is invertible and that its inverse has polynomial growth. Show that for any function ϕ ∈ Cb1 (Rd ) and for any t > 0 the following formula holds: & ' t 1 −1 j k j (A )k (Xs )Y Yi (s)dW Ws , E[∂ ∂i ϕ(Xt )] = E ϕ(Xt ) t 0 s where Y (s) denotes the Jacobian matrix ∂X ∂x0 given by (2.57). Hint: Use the decomposition Ds Xt = Y (t)Y −1 (s)A(Xs ) and the duality relationship between the derivative operator and the Skorohod (Itˆ oˆ) integral.

2.4 Stochastic partial diﬀerential equations In this section we discuss the applications of the Malliavin calculus to establishing the existence and smoothness of densities for solutions to stochastic partial diﬀerential equations. First we will treat the case of a hyperbolic system of equations using the techniques of the two-parameter stochastic calculus. Second we will prove a criterion for absolute continuity in the case of the heat equation perturbed by a space-time white noise.

2.4.1 Stochastic integral equations on the plane Suppose that W = {W Wz = (W Wz1 , . . . , Wzd ), z ∈ R2+ } is a d-dimensional, two-parameter Wiener process. That is, W is a d-dimensional, zero-mean

2.4 Stochastic partial diﬀerential equations

143

Gaussian process with a covariance function given by E[W i (s1 , t1 )W j (s2 , t2 )] = δ ij (s1 ∧ s2 )(t1 ∧ t2 ). We will assume that this process is deﬁned in the canonical probability space (Ω, F, P ), where Ω is the space of all continuous functions ω : R2+ → Rd vanishing on the axes, and endowed with the topology of the uniform convergence on compact sets, P is the law of the process W (which is called the two-parameter, d-dimensional Wiener measure), and F is the completion of the Borel σ-ﬁeld of Ω with respect to P . We will denote by {F Fz , z ∈ R2+ } the increasing family of σ-ﬁelds such that for any z, Fz is generated by the random variables {W (r), r ≤ z} and the null sets of F. Here r ≤ z stands for r1 ≤ z1 and r2 ≤ z2 . Given a rectangle ∆ = (s1 , s2 ] × (t1 , t2 ], we will denote by W (∆) the increment of W on ∆ deﬁned by W (∆) = W (s2 , t2 ) − W (s2 , t1 ) − W (s1 , t2 ) + W (s1 , t1 ). The Gaussian subspace of L2 (Ω, F, P ) generated by W is isomorphic to the Hilbert space H = L2 (R2+ ; Rd ). More precisely, to any element h ∈ H

d we associate the random variable W (h) = j=1 R2 hj (z)dW j (z). +

A stochastic process {Y (z), z ∈ R2+ } is said to be adapted if Y (z) is Fz -measurable for any z ∈ R2+ . The Itoˆ stochastic integral of adapted and square integrable processes can be constructed as in the one-parameter case and is a special case of the Skorohod integral: Proposition 2.4.1 Let L2a (R2+ × Ω) be the space of square integrable and adapted processes {Y (z), z ∈ R2+ } such that R2 E(Y 2 (z))dz < ∞. For any +

j = 1, . . . , d there is a linear isometry I j : L2a (R2+ × Ω) → L2 (Ω) such that I j (1(z1 ,z2 ] ) = W j ((z1 , z2 ]) for any z1 ≤ z2 . Furthermore, L2a (R2+ × Ω; Rd ) ⊂ Dom δ, and δ restricted to ˆ integrals I j , in the sense L2a (R2+ × Ω; Rd ) coincides with the sum of the Itˆ 2 2 that for any d-dimensional process Y ∈ La (R+ × Ω; Rd ) we have δ(Y ) =

d

I j (Y j ) =

j=1

d j=1

Y j (z)dW j (z).

R2+

Let Aj , B : Rm → Rm , 1 ≤ j ≤ d, be globally Lipschitz functions. We denote by X = {X(z), z ∈ R2+ } the m-dimensional, two-parameter, continuous adapted process given by the following system of stochastic integral equations on the plane: X(z) = x0 +

d j=1

[0,z]

Aj (Xr )dW Wrj

+

B(Xr )dr, [0,z]

(2.73)

144

2. Regularity of probability laws

where x0 ∈ Rm represents the constant value of the process X(z) on the axes. As in the one-parameter case, we can prove that this system of stochastic integral equations has a unique continuous solution: Theorem 2.4.1 There is a unique m-dimensional, continuous, and adapted process X that satisﬁes the integral equation (2.73). Moreover, )

( sup |Xr |

p

E

<∞

r∈[0,z]

for any p ≥ 2, and any z ∈ R2+ . Proof: Use the Picard iteration method and two-parameter martingale inequalities (see (A.7) and (A.8)) in order to show the uniform convergence of the approximating sequence. Equation (2.73) is the integral version of the following nonlinear hyperbolic stochastic partial diﬀerential equation: ∂ 2 W j (s, t) ∂ 2 X(s, t) = + B(X(s, t)). Aj (X(s, t)) ∂s∂t ∂s∂t j=1 d

Suppose that z = (s, t) is a ﬁxed point in R2+ not on the axes. Then we may look for nondegeneracy conditions on the coeﬃcients of Eq. (2.73) so that the random vector X(z) = (X 1 (z), . . . , X m (z)) has an absolutely continuous distribution with a smooth density. We will assume that the coeﬃcients Aj and B are inﬁnitely diﬀerentiable functions with bounded partial derivatives of all orders. We can show as in the one-parameter case that X i (z) ∈ D∞ for all z ∈ R2+ and i = 1, . . . , m. i j Furthermore, the Malliavin matrix Qij z = DXz , DXz H is given by Qij z =

d l=1

Drl Xzi Drl Xzj dr,

(2.74)

[0,z]

where for any r, the process {Drk Xzi , r ≤ z, 1 ≤ i ≤ m, 1 ≤ k ≤ d} satisﬁes the following system of stochastic diﬀerential equations: j i i ∂k Ail (Xu )Drj Xuk dW Wul Dr Xz = Aj (Xr ) + [r,z ] + ∂k B i (Xu )Drj Xuk du. (2.75) [r,z ]

Moreover, we can write Drj Xzi = ξ il (r, z)Alj (Xr ), where for any r, the process {ξ ij (r, z), r ≤ z, 1 ≤ i, j ≤ m} is the solution to the following

2.4 Stochastic partial diﬀerential equations

system of stochastic diﬀerential equations: ξ ij (r, z) = δ ij + ∂k Ail (Xu )ξ kj (r, u)dW Wul [r,z ] + ∂k B i (Xu )ξ kj (r, u)du.

145

(2.76)

[r,z ]

However, unlike the one-parameter case, the processes Drj Xzi and ξ ij (r, z) cannot be factorized as the product of a function of z and a function of r. Furthermore, these processes satisfy two-parameter linear stochastic differential equations and the solution to such equations, even in the case of constant coeﬃcients, are not exponentials, and may take negative values. As a consequence, we cannot estimate expectations such as E(|ξ ij (r, z)|−p ). The behavior of solutions to two-parameter linear stochastic diﬀerential equations is analyzed in the following proposition (cf. Nualart [243]). Proposition 2.4.2 Let {X(z), z ∈ R2+ } be the solution to the equation Xz = 1 + aXr dW Wr , (2.77) [0,z]

where a ∈ R and {W (z), z ∈ R2+ } is a two-parameter, one-dimensional Wiener process. Then, (i) there exists an open set ∆ ⊂ R2+ such that P {Xz < 0

for all

z ∈ ∆} > 0;

(ii) E(|Xz |−1 ) = ∞ for any z out of the axes. Proof:

Let us ﬁrst consider the deterministic version of Eq. (2.77):

s

g(s, t) = 1 +

t

ag(u, v)dudv. 0

(2.78)

0

The solution to this equation is g(s, t) = f (ast), where f (x) =

∞ xn . (n!)2 n=0

√ In particular, for a > 0, g(s, t) = I0 (2 ast), where I0 is%the modiﬁed Bessel function of order zero, and for a < 0, g(s, t) = J0 (2 |a|st), where J0 is the Bessel function of order zero. Note that f (x) grows exponentially as x % % 1 tends to inﬁnity and that f (x) is equivalent to (π |x|)− 2 cos(2 |x| − π4 ) as x tends to −∞. Therefore, we can ﬁnd an open interval I = (−β, −α) with 0 < α < β such that f (x) < −δ < 0 for all x ∈ I.

146

2. Regularity of probability laws

In order to show part (i) we may suppose by symmetry that a > 0. Fix N > 0 and set ∆ = {(s, t) : αa < st < βa , 0 < s, t < N }. Then ∆ is an open set contained in the rectangle T = [0, N ]2 and such that f (−ast) < −δ for any (s, t) ∈ ∆. For any > 0 we will denote by Xz the solution to the equation Xz = 1 +

aXr dW Wr . [0,z]

By Lemma 2.1.3 the process W (s, t) = W (s, t) − st−1 has the law of a two-parameter Wiener process on T = [0, N ]2 under the probability P deﬁned by dP P 1 = exp −1 W (N, N ) − −2 N 2 . dP 2 Let Yz be the solution to the equation aY Yr dW Wr = 1 + aY Yr dW Wr − Yz = 1 + [0,z]

[0,z]

aY Yr dr.

(2.79)

[0,z]

It is not diﬃcult to check that Yz |2 ) < ∞. K = sup sup E(|Y 0<≤1 z∈T

Then, for any ≤ 1, from Eqs. (2.78) and (2.79) we deduce E

sup |Y (s, t) − f (−ast)|2 (s,t)∈T

E(|Y (x, y) − f (−axy)|2 )dxdy + a2 2 K

≤C T

for some constant C > 0. Hence, lim E ↓0

sup |Y (s, t) − f (−ast)|2

P

<0

= 0,

(s,t)∈T

and therefore {Y Yz

2 for all z ∈ ∆}

≥ P

3 sup |Y (s, t) − f (−ast)| ≤ δ

(s,t)∈∆

2 ≥ P

sup |Y (s, t) − f (−ast)| ≤ δ

3 ,

(s,t)∈T

which converges to one as tends to zero. So, there exists an 0 > 0 such that P {Y Yz < 0 for all z ∈ ∆} > 0

2.4 Stochastic partial diﬀerential equations

147

for any ≤ 0 . Then P {Y Yz < 0

for all

z ∈ ∆} > 0

because the probabilities P and P are equivalent, and this implies P {Xz < 0

for all

z ∈ ∆} > 0.

By the scaling property of the two-parameter Wiener process, the processes X (s, t) and X(s, t) have the same law. Therefore, P {X(s, t) < 0

for all (s, t) ∈ ∆} > 0,

which gives the desired result with the open set ∆ for all ≤ 0 . Note that one can also take the open set {(2 s, t) : (s, t) ∈ ∆}. To prove (ii) we ﬁx (s, t) such that st = 0 and deﬁne T = inf{σ ≥ 0 : X(σ, t) = 0}. T is a stopping time with respect to the increasing family of σ-ﬁelds {F Fσt , σ ≥ 0}. From part (i) we have P {T < s} > 0. Then, applying Itˆ o’s formula in the ﬁrst coordinate, we obtain for any > 0 1

1

E[(X(s, t)2 + )− 2 ] = E[(X(s ∧ T, t)2 + )− 2 ] & ' s 5 1 + E (2X(x, t)2 − )(X(x, t)2 + )− 2 dX(·, t)x . 2 s∧T Finally, if ↓ 0, by monotone convergence we get 1

E(|X(s, t)|−1 ) = lim E[(X(s, t)2 + )− 2 ] ≥ ∞P {T < s} = ∞. ↓0

In spite of the technical problems mentioned before, it is possible to show the absolute continuity of the random vector Xz solution of (2.73) under some nondegeneracy conditions that diﬀer from H¨ ormander’s hypothesis. We introduce the following hypothesis on the coeﬃcients Aj and B, which are assumed to be inﬁnitely diﬀerentiable with bounded partial derivatives of all orders: (P) The vector space spanned by the vector ﬁelds A1 , . . . , Ad , A∇ i Aj , ∇ Ai1 (· · · (A∇ 1 ≤ i, j ≤ d, A∇ i (Aj Ak ), 1 ≤ i, j, k ≤ d, . . . , in−1 Ain ) · · · ), 1 ≤ i1 , . . . , in ≤ d, at the point x0 is Rm . Then we have the following result.

148

2. Regularity of probability laws

Theorem 2.4.2 Assume that condition (P) holds. Then for any point z out of the axes the random vector X(z) has an absolutely continuous probability distribution. We remark that condition (P) and H¨ o¨rmander’s hypothesis (H) are not comparable. Consider, for instance, the following simple example. Assume that m ≥ 2, d = 1, x0 = 0, A1 (x) = (1, x1 , x2 , . . . , xm−1 ), and B(x) = 0. This means that Xz is the solution of the diﬀerential system dXz1 dXz2 dXz3 dXzm

= = = ··· =

dW Wz Xz1 dW Wz Xz2 dW Wz Xzm−1 dW Wz ,

and Xz = 0 if z is on the axes. Then condition (P) holds and, as a consequence, Theorem 2.4.2 implies that the joint of the iter distribution ated stochastic integrals Wz , [0,z] W dW , . . . , [0,z] (· · · ( W dW ) · · · )dW = dW (z1 ) · · · dW (zm ) possesses a density on Rm . However, H¨ormanz1 ≤···≤zm der’s hypothesis is not true in this case. Notice that in the one-parameter t Ws is case the joint distribution of the random variables Wt and 0 Ws dW t singular because Itˆo’s formula implies that Wt2 − 2 0 Ws dW Ws − t = 0. Proof of Theorem 2.4.2: The ﬁrst step will be to show that the process ξ ij (r, z) given by system (2.76) has a version that is continuous in the variable r ∈ [0, z]. By means of Kolmogorov’s criterion (see the appendix, Section A.3), it suﬃces to prove the following estimate: p

E(|ξ(r, z) − ξ(r , z)|p ) ≤ C(p, z)|r − r | 2

(2.80)

for any r, r ∈ [0, z] and p > 4. One can show that sup E r∈[0,z]

sup |ξ(r, v)|p

≤ C(p, z),

(2.81)

v∈[r,z]

where the constant C(p, z) depends on p, z and on the uniform bounds of the derivatives ∂k B i and ∂k Ail . As a consequence, using Burkholder’s and H¨ older’s inequalities, we can write

2.4 Stochastic partial diﬀerential equations

149

E(|ξ(r, z) − ξ(r , z)|p ) ⎧ ⎛ m ⎨ ∂k Ail (Xv )(ξ kj (r, v) − ξ kj (r , v))dW ≤ C(p, z) E ⎝ Wvl ⎩ [ r ∨r ,z] i,j=1 ⎞ p 2 2 ⎟ + ∂k B i (Xv )(ξ kj (r, v) − ξ kj (r , v))dv ⎠ ⎛ m +E ⎝ Wvl ∂k Ail (Xv )ξ kj (r, v)dW [r,z ]−[r ,z] i,j=1 ⎞ p 2 2 ⎟ + ∂k B i (Xv )ξ kj (r, v)dv ⎠ ⎛ m ⎝ ∂k Ail (Xv )ξ kj (r , v)dW +E Wvl [r ,z]−[r,z] i,j=1 ⎞⎫ p ⎬ 2 2 ⎪ ⎟ + ∂k B i (Xv )ξ kj (r , v)dv ⎠ ⎪ ⎭ p 2

≤ C(p, z) |r − r | +

[r ∨r ,z]

E(|ξ(r, v) − ξ(r , v)|p )dv .

Using a two-parameter version of Gronwall’s lemma (see Exercise 2.4.3) we deduce Eq. (2.80). In order to prove the theorem, it is enough to show that det Qz > 0 a.s., where z = (s, t) is a ﬁxed point such that st = 0, and Qz is given by (2.74). Suppose that P {det Qz = 0} > 0. We want to show that under this assumption condition (P) cannot be satisﬁed. For any σ ∈ (0, s] let Kσ denote the vector subspace of Rm spanned by {Aj (Xξt ); 0 ≤ ξ ≤ σ, j = 1, . . . , d}. Then {Kσ , 0 < σ ≤ s} is an increasing family of subspaces. We set K0+ = ∩σ>0 Kσ . By the Blumenthal zero-one law, K0+ is a deterministic subspace with probability one. Deﬁne ρ = inf{σ > 0 : dim Kσ > dim K0+ }. Then ρ > 0 a.s., and ρ is a stopping time with respect to the increasing family of σ-ﬁelds {F Fσt , σ ≥ 0}. For any vector v ∈ Rm we have d T v Qz v = (vi ξ il (r, z)Alj (Xr ))2 dr. j=1

[0,z]

150

2. Regularity of probability laws

Assume that v T Qz v = 0. Due to the continuity in r of ξ ij (r, z), we deduce vi ξ il (r, z)Alj (Xr ) = 0 for any r ∈ [0, z] and for any j = 1, . . . , d. In particular, for r = (σ, t) we get v T Aj (Xσt ) = 0 for any σ ∈ [0, s]. As a consequence, K0+ = Rm . Otherwise Kσ = Rm for all σ ∈ [0, s], and any vector v verifying v T Qz v = 0 would be equal to zero. So, Qz would be invertible a.s., which contradicts our assumption. Let v be a ﬁxed nonzero vector orthogonal to K0+ . We remark that v is orthogonal to Kσ if σ < ρ, that is, v T Aj (Xσt ) = 0

for all σ < ρ

and j = 1, . . . , d.

(2.82)

We introduce the following sets of vector ﬁelds: Σ0 = {A1 , . . . , Ad }, Σn = {A∇ j V, j = 1, . . . , d, V ∈ Σn−1 }, Σ = ∪∞ n=0 Σn .

n ≥ 1,

Under property (P), the vector space Σ(x0 ) spanned by the vector ﬁelds of Σ at point x0 has dimension m. We will show that the vector v is orthogonal to Σn (x0 ) for all n ≥ 0, which contradicts property (P). Actually, we will prove the following stronger orthogonality property: v T V (Xσt ) = 0 for all σ < ρ, V ∈ Σn

and n ≥ 0.

(2.83)

Assertion (2.83) is proved by induction on n. For n = 0 it reduces to (2.82). Suppose that it holds for n − 1, and let V ∈ Σn−1 . The process {v T V (Xσt ), σ ∈ [0, s]} is a continuous semimartingale with the following integral representation: v T V (Xσt ) = v T V (x0 ) + 0

σ

t 0

j v T (∂k V )(Xξt )Akj (Xξτ )dW Wξτ

+ v T (∂k V )(Xξt )B k (Xξτ )dξdτ

) d 1 T k k + v ∂k ∂k V (Xξt ) Al (Xξτ )Al (Xξτ )dξdτ . 2 l=1

The quadratic variation of this semimartingale is equal to d j=1

0

σ

t

2 v T (∂k V )(Xξt )Akj (Xξτ ) dξdτ .

0

By the induction hypothesis, the semimartingale vanishes in the random interval [0, ρ). As a consequence, its quadratic variation is also equal to zero in this interval, and we have, in particular,

2.4 Stochastic partial diﬀerential equations

v T (A∇ j V )(Xσt ) = 0 for all σ < ρ

and

151

j = 1, . . . , d.

Thus, (2.83) holds for n. This achieves the proof of the theorem.

It can be proved (cf. [256]) that under condition (P), the density of Xz is inﬁnitely diﬀerentiable. Moreover, it is possible to show the smoothness of the density of Xz under assumptions that are weaker than condition (P). In fact, one can consider the vector space spanned by the algebra generated by A1 , . . . , Ad with respect to the operation ∇ , and we can also add other generators formed with the vector ﬁeld B. We refer to references [241] and [257] for a discussion of these generalizations.

2.4.2 Absolute continuity for solutions to the stochastic heat equation Suppose that W = {W (t, x), t ∈ [0, T ], x ∈ [0, 1]} is a two-parameter Wiener process deﬁned on a complete probability space (Ω, F, P ). For each t ∈ [0, T ] we will denote by Ft the σ-ﬁeld generated by the random variables {W (s, x), (s, x) ∈ [0, t] × [0, 1]} and the P -null sets. We say that a random ﬁeld {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} is adapted if for all (t, x) the random variable u(t, x) is Ft -measurable. Consider the following parabolic stochastic partial diﬀerential equation on [0, T ] × [0, 1]: ∂2u ∂2W ∂u = + b(u(t, x)) + σ(u(t, x)) 2 ∂t ∂x ∂t∂x

(2.84)

with initial condition u(0, x) = u0 (x), and Dirichlet boundary conditions u(t, 0) = u(t, 1) = 0. We will assume that u0 ∈ C([0, 1]) satisﬁes u0 (0) = u0 (1) = 0. It is well known that the associated homogeneous equation (i.e., when b ≡ 1 0 and σ ≡ 0) has a unique solution given by v(t, x) = 0 Gt (x, y)u0 (y)dy, where Gt (x, y) is the fundamental solution of the heat equation with Dirichlet boundary conditions. The kernel Gt (x, y) has the following explicit formula: ∞ 1 (y − x − 2n)2 exp − Gt (x, y) = √ 4t 4πt n=−∞ (y + x − 2n)2 − exp − . (2.85) 4t On the other hand, Gt (x, y) coincides with √ the probability density at point y of a Brownian motion with variance 2t starting at x and killed if it leaves the iterval [0, 1]. This implies that |x − y|2 1 exp − Gt (x, y) ≤ √ . (2.86) 4t 4πt

152

2. Regularity of probability laws

Therefore, for any β > 0 we have

1

β

Gt (x, y)β dy ≤ (4πt)− 2

e−

β|x|2 4t

R

0

dx = Cβ t

1−β 2

.

(2.87)

Note that the right-hand side of (2.87) is integrable in t near the origin, provided that β < 3. 2 W does not exist, and Equation (2.84) is formal because the derivative ∂∂t∂x we will replace it by the following integral equation: u(t, x)

t

1

1

Gt (x, y)u0 (y)dy +

=

Gt−s (x, y)b(u(s, y))dyds

0

0

t

0

1

+

Gt−s (x, y)σ(u(s, y))W (dy, ds) . 0

(2.88)

0

One can deﬁne a solution to (2.84) in terms of distributions and then show that such a solution exists if and only if (2.88) holds. We refer to Walsh [342] for a detailed discussion of this topic. We can state the following result on the integral equation (2.88). Theorem 2.4.3 Suppose that the coeﬃcients b and σ are globally Lipschitz functions. Then there is a unique adapted process u = {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} such that

T

1

2

u(t, x) dxdt

E 0

< ∞,

0

and satisﬁes (2.88). Moreover, the solution u satisﬁes E(|u(t, x)|p ) < ∞

sup

(2.89)

(t,x)∈[0,T ]×[0,1]

for all p ≥ 2. Proof:

Consider the Picard iteration scheme deﬁned by

1

u0 (t, x) =

Gt (x, y)u0 (y)dy 0

and t un+1 (t, x)

= u0 (t, x) + t +

1

Gt−s (x, y)b(un (s, y))dyds 0

0

1

Gt−s (x, y)σ(un (s, y))W (dy, ds), 0

0

(2.90)

2.4 Stochastic partial diﬀerential equations

153

n ≥ 0. Using the Lipschitz condition on b and σ and the isometry property of the stochastic integral with respect to the two-parameter Wiener process (see the Appendix, Section A.3), we obtain E(|un+1 (t, x) − un (t, x)|2 ) 2 t 1 ≤ 2E Gt−s (x, y)|un (s, y) − un−1 (s, y)|dyds 0

t

0

Gt−s (x, y) |un (s, y) − un−1 (s, y)| dydss

1

2

+2E 0

0

t

1

≤ 2(T + 1) 0

2

Gt−s (x, y)2 E |un (s, y) − un−1 (s, y)|2 dyds.

0

Now we apply (2.87) with β = 2, and we obtain E(|un+1 (t, x) − un (t, x)|2 ) t 1 1 E(|un (s, y) − un−1 (s, y)|2 )(t − s)− 2 dyds. ≤ CT 0

0

Hence, E(|un+1 (t, x) − un (t, x)|2 ) t s 1 1 1 E(|un (r, z) − un−1 (r, z)|2 )(s − r)− 2 (t − s)− 2 dzdrds ≤ CT2 0

=

CT

0

t 0

0 1

E(|un (r, z) − un−1 (r, z)|2 )dzdr.

0

Iterating this inequality yields 1 ∞ sup E(|un+1 (t, x) − un (t, x)|2 )dx < ∞. n=0 t∈[0,T ]

0

This implies that the sequence un (t, x) converges in L2 ([0, 1]×Ω), uniformly in time, to a stochastic process u(t, x). The process u(t, x) is adapted and satisﬁes (2.88). Uniqueness is proved by the same argument. Let us now show (2.89). Fix p > 6. Applying Burkholder’s inequality for stochastic integrals with respect to the Brownian sheet (see (A.8)) and the boundedness of the function u0 yields p

E (|un+1 (t, x)| )

≤

p

cp (u0 ∞ t +E 0

t

1

p Gt−s (x, y) |b(un (s, y))| dyds

0

+E

1

2

2

Gt−s (x, y) σ(un (s, y)) dyds 0

0

p2 .

154

2. Regularity of probability laws

Using the linear growth condition of b and σ we can write p2 t 1 p 2 2 Gt−s (x, y) un (s, y) dyds . E (|un+1 (t, x)| ) ≤ Cp,T 1 + E 0

0

2p Now we apply Holder’s ¨ inequality and (2.87) with β = p−2 < 3, and we obtain ⎛ p−2 t 1 2 2p p ⎝ p−2 Gt−s (x, y) dyds E (|un+1 (t, x)| ) ≤ Cp,T 1 + 0

t

1

×

p

E(|un (s, y)| )dyds 0

≤

0

Cp,T

0

t 1+ 0

1

p

E(|un (s, y)| )dyds ,

0

and we conclude using Gronwall’s lemma. The next proposition tells us that the trajectories of the solution to the Equation (2.88) are α-Holder ¨ continuous for any α < 14 . For its proof we need the following technical inequalities. (a) Let β ∈ (1, 3). For any x ∈ [0, 1] and t, h ∈ [0, T ] we have t 1 3−β |Gs+h (x, y) − Gs (x, y)|β dyds ≤ CT,β h 2 , 0

(b) Let β ∈ ( 32 , 3). For any x, y ∈ [0, 1] and t ∈ [0, T ] we have t 1 |Gs (x, z) − Gs (y, z)|β dzds ≤ CT,β |x − y|3−β . 0

(2.91)

0

(2.92)

0

H¨ continuous function Proposition 2.4.3 Fix α < 14 . Let u0 be a 2α-Holder such that u0 (0) = u0 (1) = 0. Then, the solution u to Equation (2.88) has a version with α-Holder ¨ continuous paths. Proof: We ﬁrst check the regularity of the ﬁrst term in (2.88). Set 1 Gt (x, u0 ) := 0 Gt (x, y)u0 (y)dy. The semigroup property of G implies 1 1 Gt (x, u0 ) − Gs (x, u0 ) = Gs (x, y)Gt−s (y, z)[u0 (z) − u0 (y)]dzdy. 0

0

Hence, using (2.86) we get |Gt (x, u0 ) − Gs (x, u0 )|

1

≤ C

0

≤ C

0

1

Gs (x, y)Gt−s (y, z)|z − y|2α dzdy

0 1

Gs (x, y)|t − s|α dy ≤ C |t − s|α .

2.4 Stochastic partial diﬀerential equations

155

On the other hand, from (2.85) we can write Gt (x, y) = ψ t (y − x) − ψ t (y + x), 1

+∞ 1 −(x−2n)/4t where ψ t (x) = √4πt . Notice that supx∈[0,1] 0 ψ t (z − n=−∞ e x)dz ≤ C. We can write Gt (x, u0 ) − Gt (y, u0 )

1

=

[ψ t (z − x) − ψ t (z − y)] u0 (z)dz

0

1

−

[ψ t (z + x) − ψ t (z + y)] u0 (z)dz

0

= A1 + B1 . It suﬃces to consider the term A1 , because B1 can be treated by a similar method. Let η = y − x > 0. Then, using the H¨ o¨lder continuity of u0 and the fact that u0 (0) = u1 (0) = 1 we obtain

1−η

|A1 | ≤

ψ t (z − x) |u0 (z) − u0 (z + η)| dz

0

1

+ 1−η

≤ Cη

2α

ψ t (z − x) |u0 (z)| dz +

η

ψ t (z − y) |u0 (z)| dz 0

1

ψ t (z − x)(1 − z) dz + C

+C 1−η

η

ψ t (z − y)z 2α dz

2α

0

≤ C η 2α . Set

t U (t, x) =

1

Gt−s (x, y)σ(u(s, y))W (dy, ds). 0

0

Applying Burkholder’s and Holder’s ¨ inequalities (see (A.8)), we have for any p > 6 E(|U (t, x) − U (t, y)|p ) p2 t 1 2 2 |Gt−s (x, z) − Gt−s (y, z)| |σ(u(s, z))| dzds ≤ Cp E 0

0

t ≤ Cp,T 0

because

T 1 0

0

1

|Gt−s (x, z) − Gt−s (y, z)|

2p p−2

p−2 2 dzds

,

0

E(|σ(u(s, z))|p )dzds < ∞. From (2.92) with β =

know that this is bounded by C|x − y|

p−6 2

.

2p p−2 ,

we

156

2. Regularity of probability laws

On the other hand, for t > s we can write E(|U (t, x) − U (s, x)|p ) 2 p2 s 1 2 2 |Gt−θ (x, y) − Gs−θ (x, y)| |σ(u(θ, y))| dydθ ≤ Cp E 0 0 p2 3 t 1 2 2 + E |Gt−θ (x, y)| |σ(u(θ, y))| dydθ s 0 ⎧ p−2 ⎨ s 1 2 2p ≤ Cp,T |Gt−θ (x, y) − Gs−θ (x, y)| p−2 dydθ ⎩ 0

+

0

s

0

1

Gt−θ (x, y)

2p p−2

0

⎫ p−2 2 ⎬ dydθ . ⎭

p−6

Using (2.91) we can bound the ﬁrst summand by Cp |t − s| 4 . From (2.87) the second summand is bounded by t−s 1 t−s p+2 2p p−2 Gθ (x, y) dydθ ≤ Cp θ− 2(p−2) dθ 0

0

0

=

Cp |t

p−6

− s| 2(p−2) .

As a consequence,

p−6 p−6 E(|U (t, x) − U (s, y)|p ) ≤ Cp,T |x − y| 2 + |t − s| 4 ,

and we conclude using Kolmogorov’s continuity criterion. In a similar way we can handle that the term t 1 V (t, x) = Gt−s (x, y)b(u(s, y))dyds. 0

0

In order to apply the criterion for absolute continuity, we will ﬁrst show that the random variable u(t, x) belongs to the space D1,2 . Proposition 2.4.4 Let b and σ be Lipschitz functions. Then u(t, x) ∈ D1,2 , and the derivative Ds,y u(t, x) satisﬁes Ds,y u(t, x)

= Gt−s (x, y)σ(u(s, y)) t 1 + Gt−θ (x, η)Bθ,η Ds,y u(θ, η)dηdθ s

0

t +

1

Gt−θ (x, η)Sθ,η Ds,y u(θ, η)W (dθ, dη) s

0

if s < t, and Ds,y u(t, x) = 0 if s > t, where Bθ,η and Sθ,η , (θ, η) ∈ [0, T ] × [0, 1], are adapted and bounded processes.

2.4 Stochastic partial diﬀerential equations

157

Remarks: If the coeﬃcients b and σ are functions of class C 1 with bounded derivatives, then Bθ,η = b (u(θ, η)) and Sθ,η = σ (u(θ, η)). Proof: Consider the Picard approximations un (t, x) introduced in (2.90). Suppose that un (t, x) ∈ D1,2 for all (t, x) ∈ [0, T ] × [0, 1] and t 1 2 E |Ds,y un (t, x)| dyds < ∞. (2.93) sup (t,x)∈[0,T ]×[0,1]

0

0

Applying the operator D to Eq. (2.90), we obtain that un+1 (t, x) ∈ D1,2 and that Ds,y un+1 (t, x)

= Gt−s (x, y)σ(un (s, y)) t 1 n + Gt−θ (x, η)Bθ,η Ds,y un (θ, η)dηdθ 0

s

t

1

n Gt−θ (x, η)Sθ,η Ds,y un (θ, η)W (dθ, dη),

+ 0

s

n n where Bθ,η and Sθ,η , (θ, η) ∈ [0, T ]×[0, 1], are adapted processes, uniformly bounded by the Lipschitz constants of b and σ, respectively. Note that T 1 2 2 Gt−s (x, y) σ(un (s, y)) dyds E 0

0

≤ C1

1+

E(un (t, x)2 )

sup t∈[0,T ],x∈[0,1]

≤ C2 ,

for some constants C1 , C2 > 0. Hence t 1 2 |Ds,y un+1 (t, x)| dyds E 0

0

t 1 t 1 ≤ C3 1 + E Gt−θ (x, η)2 |Ds,y un (θ, η)|2 dηdθdyds 0 0 s 0 t

≤ C4

1+

1

t

sup 0 η∈[0,1]

1

(t − θ)− 2 E(|Ds,y un (θ, η)|2 )dθdyds .

0

s

Let

t Vn (t) = sup E x∈[0,1]

Then Vn+1 (t)

0

1

|Ds,y un (t, x)|2 dyds .

0

t 1 ≤ C4 1 + Vn (θ)(t − θ)− 2 dθ 0 t

≤ C5

θ

− 12

Vn−1 (u)(t − θ)

1+ 0

0

t ≤ C6 1 + Vn−1 (u)du < ∞, 0

− 12

(θ − u)

dudθ

158

2. Regularity of probability laws

due to (2.93). By iteration this implies that Vn (t) < C,

sup t∈[0,T ],x∈[0,1]

where the constant C does not depend on n. Taking into account that un (t, x) converges to u(t, x) in Lp (Ω) for all p ≥ 1, we deduce that u(t, x) ∈ D1,2 , and Dun (t, x) converges to Du(t, x) in the weak topology of L2 (Ω; H) (see Lemma 1.2.3). Finally, applying the operator D to both members of Eq. (2.88), we deduce the desired result. The main result of this section is the following; Theorem 2.4.4 Let b and σ be globally Lipschitz functions. Assume that σ(u0 (y)) = 0 for some y ∈ (0, 1). Then the law of u(t, x) is absolutely continuous for any (t, x) ∈ (0, T ] × (0, 1). Proof: Fix (t, x) ∈ (0, T ] × (0, 1). According to the general criterion for absolute continuity (Theorem 2.1.3), we have to show that t 1 |Ds,y u(t, x)|2 dyds > 0 (2.94) 0

0

a.s. There exists an interval [a, b] ⊂ (0, 1) and a stopping time τ > 0 such that σ(u(s, y)) ≥ δ > 0 for all y ∈ [a, b] and 0 ≤ s ≤ τ . Then a suﬃcient condition for (2.94) is b Ds,y u(t, x)dy > 0 for all 0 ≤ s ≤ τ , (2.95) a

a.s. for some b ≥ a. We will show (2.95) only for the case where s = 0. The case where s > 0 can be treated by similar arguments, restricting the study to the set {s < τ }. On the other hand, one can show using Kolmogorov’s continuity criterion that the process {Ds,y u(t, x), s ∈ [0, t], y ∈ [0, 1]} possesses a continuous version, and this implies that it suﬃces to consider the case s = 0. The process b D0,y u(t, x)dy v(t, x) = a

is the unique solution of the following linear stochastic parabolic equation: t 1 b Gt (x, y)σ(u0 (y))dy + Gt−s (x, y)Bs,y v(s, y)dsdy v(t, x) = 0

a

t

0

1

Gt−s (x, y)Ss,y v(s, y)W (ds, dy).

+ 0

(2.96)

0

We are going to prove that the solution to this equation is strictly positive at (t, x). By the comparison theorem for stochastic parabolic equations (see

2.4 Stochastic partial diﬀerential equations

159

Exercise 2.4.5) it suﬃces to show the result when the initial condition is δ1[a,b] , and by linearity we can take δ = 1. Moreover, for any constant c > 0 the process ect v(t, x) satisﬁes the same equation as v but with Bs,y replaced by Bs,y + c. Hence, we can assume that Bs,y ≥ 0, and by the comparison theorem it suﬃces to prove the result with B ≡ 0. Suppose that a ≤ x < 1 (the case where 0 < x ≤ a would be treated by similar arguments). Let d > 0 be such that x ≤ b + d < 1. We divide [0, t] kt into m smaller intervals [ k−1 m t, m ], 1 ≤ k ≤ m. We also enlarge the interval [a, b] at each stage k, until by stage k = m it covers [a, b + d]. Set 1 α = inf inf inf 2 m≥1 1≤k≤m y∈[a,b+ kd m]

b+

d(k−1) m

a

G mt (y, z)dz,

and note that α > 0. For k = 1, 2, . . . , m we deﬁne the set Ek =

kt v( , y) ≥ αk 1[a,b+ kd ] (y), ∀y ∈ [0, 1] . m m

We claim that for any δ > 0 there exists m0 ≥ 1 such that if m ≥ m0 then c |E1 ∩ · · · ∩ Ek ) ≤ P (Ek+1

δ m

(2.97)

for all 0 ≤ k ≤ m − 1. If this is true, then we obtain P {v(t, x) > 0}

9 : ≥ P v(t, y) ≥ αm 1[a,b+d] (y), ∀y ∈ [0, 1] ≥ P (Em |Em−1 ∩ · · · ∩ E1 ) ×P (Em−1 |Em−2 ∩ · · · ∩ E1 ) . . . P (E1 ) m δ ≥ 1 − δ, ≥ 1− m

and since δ is arbitrary we get P {v(t, x) > 0} = 1. So it only remains to t(k+1) check Eq. (2.97). We have for s ∈ [ tk m, m ] v(s, y)

1

=

G mt (y, z)v(

0

s

kt , z)dz m

1

Gs−θ (y, z)Sθ,z v(θ, z)W (dθ, dz).

+ t m

0

Again by the comparison theorem (see Exercise 2.4.5) we deduce that on the set E1 ∩ · · · ∩ Ek the following inequalities hold v(s, y) ≥ w(s, y) ≥ 0

160

2. Regularity of probability laws

t(k+1) tk t(k+1) for all (s, y) ∈ [ tk m, m ] × [0, 1], where w = {w(s, y), (s, y) ∈ [ m , m ]× [0, 1]} is the solution to

w(s, y)

1

= 0

s

G mt (y, z)αk 1[a,b+ kd ] (z)dz

m

1

+

Gs−θ (y, z)Sθ,z w(θ, z)W (dθ, dz). k m

0

Hence, P (Ek+1 |E1 ∩ · · · ∩ Ek ) (k + 1)t (k + 1)d ≥ P w( , y) ≥ αk+1 , ∀y ∈ [a, b + ] . (2.98) m m On the set Ek and for y ∈ [a, b +

b+ kd m

a

(k+1)d m ],

it holds that

G mt (y, z)dz ≥ 2α.

Thus, from (2.98) we obtain that ⎛

⎞

c |E1 ∩ · · · ∩ Ek ) ≤ P ⎝ P (Ek+1

sup

(k+1)d y∈[a,b+ m ]

|Φk+1 (y)| > α|E1 ∩ · · · ∩ Ek ⎠

≤ α

−p

sup |Φk+1 (y)| |E1 ∩ · · · ∩ Ek p

E

,

y∈[0,1]

for any p ≥ 2, where Φk+1 (y) =

t(k+1) m k m

0

1

G t(k+1) −s (y, z)Ss,z m

w(s, z) W (ds, dz). αk

Applying Burkholder’s inequality and taking into account that Ss,z is uniformly bounded we obtain E (|Φk+1 (y1 ) − Φk+1 (y2 )|p |E1 ∩ · · · ∩ Ek ) t 1 m ≤ CE (Gs (y1 , z) − Gs (y2 , z))2 α−2k 0 0 p2 2 t(k + 1) − s, z) dsdz |E1 ∩ · · · ∩ Ek . w( m Note that supk≥1,z∈[0,1],s∈[ tk , t(k+1) ] α−2kq E w(s, z)2q |E1 ∩ · · · ∩ Ek is m m bounded by a constant not depending on m for all q ≥ 2. As a conse-

2.4 Stochastic partial diﬀerential equations

161

quence, Holder’s ¨ inequality and Eq. (2.68) yield for p > 6 E (|Φk+1 (y1 ) − Φk+1 (y2 )|p |E1 ∩ · · · ∩ Ek ) p 3η η1 t 1 m t 3η |Gs (y1 , z) − Gs (y2 , z)| dsdz ≤ C m 0 0 1

≤ Cm− η |x − y| where

2 3

∨

2 p

p(1−η) η

,

< η < 1. Now from (A.11) we get E

sup |Φk+1 (y)|p |E1 ∩ · · · ∩ Ek

1

≤ Cm− η ,

y∈[0,1]

which concludes the proof of (2.97).

Exercises 2.4.1 Prove Proposition 2.4.1. Hint: Use the same method as in the proof of Proposition 1.3.11. 2.4.2 Let {Xz , z ∈ R2+ } be the two-parameter process solution to the linear equation Xz = 1 +

aXr dW Wr . [0,z]

Find the Wiener chaos expansion of Xz . 2.4.3 Let α, β : R2+ → R be two measurable and bounded functions. Let f : R2+ → R be the solution of the linear equation f (z) = α(z) + β(r)f (r)dr. [0,z]

Show that for any z = (s, t) we have |f (z)| ≤ sup |α(r)| r∈[0,z]

∞ m=0

(m!)−2 sup |β(r)|m (st)m . r∈[0,z]

2.4.4 Prove Eqs. (2.91) and (2.92). |x−y|2

1 Hint: It suﬃces to consider the term √4πt e− 4t in the series expansion of Gt (x, y). Then, for the proof of (2.92) it is convenient to majorize by the integral over [0, t] × R and make the change of variables z = (x −√y)ξ, s = (x − y)2 η. For (2.91) use the change of variables s = hu and y = hz.

2.4.5 Consider the pair of parabolic stochastic partial diﬀerential equations ∂ui ∂ 2 ui ∂2W i i = , + f (u (t, x))B(t, x) + g(u (t, x))G(t, x) i ∂t ∂x2 ∂t∂x

i = 1, 2,

162

2. Regularity of probability laws

where fi , g are Lipschitz functions, and B and G are measurable, adapted, and bounded random ﬁelds. The initial conditions are ui (0, x) = ϕi (x). Then ϕ1 ≤ ϕ2 (f1 ≤ f2 ) implies u1 ≤ u2 . Hint: Let {ei , i ≥ 1} be a complete orthonormal system on L2 ([0, 1]). Projecting the above equations on the ﬁrst N vectors produces a stochastic partial diﬀerential equation driven by the N independent Brownian motions deﬁned by 1

W i (t) =

ei (x)W (t, dx),

i = 1, . . . , N.

0

In this case we can use Ito’s ˆ formula to get the inequality, and in the general case one uses a limit argument (see Donati-Martin and Pardoux [83] for the details). 2.4.6 Let u = {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} be an adapted process such T 1 that 0 0 E(u2s,y )dyds < ∞. Set t

1

Zt,x =

Gt−s (x, y)us,y dW Ws,y . 0

0

Show the following maximal inequality p E sup |Z Zt,x | 0≤t≤T

T

≤ Cp,T

t

1

1

E 0

0

0

where α < 14 and p > Hint: Write Zt,x =

s)−2α u2s,y dyds

p2 dxdt,

0

3 2α .

sin πα π

where

Gt−s (x, y) (t − 2

s

t 0

Ys,y = 0

1

1

Gt−s (x, y)(t − s)α−1 Ys,y dyds,

0

Gs−θ (y, z)(s − θ)−α uθ,z dW Wθ,z ,

0

and apply H¨ o¨lder and Burholder’s inequalities.

Notes and comments [2.1] The use of the integration-by-parts formula to deduce the existence and regularity of densities is one of the basic applications of the Malliavin calculus, and it has been extensively developed in the literature. The starting point of these applications was the paper by Malliavin [207] that exhibits a probabilistic proof of H¨ o¨rmander’s theorem. Stroock

2.4 Stochastic partial diﬀerential equations

163

[318], Bismut [38], Watanabe [343], and others, have further developed the technique Malliavin introduced. The absolute continuity result stated in Theorem 2.1.1 is based on Shigekawa’s paper [307]. Bouleau and Hirsch [46] introduced an alternative technique to deal with the problem of the absolute continuity, and we described their approach in Section 2.1.2. The method of Bouleau and Hirsch works in the more general context of a Dirichlet form, and we refer to reference [47] for a complete discussion of this generalization. The simple proof of Bouleau and Hirsch criterion’s for absolute continuity in dimension one stated in Theorem 2.1.3 is based on reference [266]. For another proof of a similar criterion of absolute continuity, we refer to the note of Davydov [77]. The approach to the smoothness of the density based on the notion of distribution on the Wiener space was developed by Watanabe [343] and [144]. The main ingredient in this approach is the fact that the composition of a Schwartz distribution with a nondegenerate random vector is well deﬁned as a distribution on the Wiener space (i.e., as an element of D−∞ ). Then we can interpret the density p(x) of a nondegenerate random vector F as the expectation E[δ x (F )], and from this representation we can deduce that p(x) is inﬁnitely diﬀerentiable. The connected property of the topological support of the law of a smooth random variable was ﬁrst proved by Fang in [95]. For further works on the properties on the positivity of the density of a random vector we refer to [63]. On the other hand, general criterion on the positivity of the density using technique of Malliavin calculus can be deduced (see [248]). The fact that the supremum of a continuous process belongs to D1,2 (Proposition 2.1.10) has been proved in [261]. Another approach to the diﬀerentiability of the supremum based on the derivative of Banach-valued functionals is provided by Bouleau and Hirsch in [47]. The smoothness of the density of the Wiener sheet’s supremum has been established in [107]. By a similar argument one can show that the supremum of the fractional Brownian motion has a smooth density in (0, +∞) (see [190]). In the case of a Gaussian process parametrized by a compact metric space S, Ylvisaker [352], [353] has proved by a direct argument that the supremum has a bounded density provided the variance of the process is equal to 1. See also [351, Theorem 2.1]. [2.2] The weak diﬀerentiabilility of solutions to stochastic diﬀerential equations with smooth coeﬃcients can be proved by several arguments. In [146] Ikeda and Watanabe use the approximation of the Wiener process by means of polygonal paths. They obtain a sequence of ﬁnite-diﬀerence equations whose solutions are smooth functionals that converge to the diﬀusion process in the topology of D∞ . Stroock’s approach in [320] uses an iterative family of Hilbert-valued stochastic diﬀerential equations. We have used the Picard iteration scheme Xn (t). In order to show that the limit X(t) belongs to the space D∞ , it suﬃces to show the convergence in Lp , for any

164

2. Regularity of probability laws

p ≥ 2, and the boundedness of the derivatives DN Xn (t) in Lp (Ω; H ⊗N ), uniformly in n. In the one-dimensional case, Doss [84] has proved that a stochastic diﬀerential equation can be solved path-wise – it can be reduced to an ordinary diﬀerential equation (see Exercise 2.2.2). This implies that the solution in this case is not only in the space D1,p but, assuming the coeﬃcients are of ´echet diﬀerentiable on the Wiener space C0 ([0, T ]). class C 1 (R), that it is Fr´ In the multidimensional case the solution might not be a continuous functional of the Wiener process. The simplest example of this situation is Levy’s ´ area (cf. Watanabe [343]). However, it is possible to show, at least if ¨ ¨ the coeﬃcients have compact support (Ustunel and Zakai [337]), that the solution is H-continuously diﬀerentiable. The notion of H-continuous differentiability will be introduced in Chapter 4 and it requires the existence and continuity of the derivative along the directions of the Cameron-Martin space. [2.3] The proof of H¨ o¨rmander’s theorem using probabilistic methods was ﬁrst done by Malliavin in [207]. Diﬀerent approaches were developed after Malliavin’s work. In [38] Bismut introduces a direct method for proving H¨ o¨rmander’s theorem, based on integration by parts on the Wiener space. Stroock [319, 320] developed the Malliavin calculus in the context of a symmetric diﬀusion semigroup, and a general criteria for regularity of densities was provided by Ikeda and Watanabe [144, 343]. The proof we present in this section has been inspired by the work of Norris [239]. The main ingredient is an estimation for continuous semimartingales (Lemma 2.3.2), which was ﬁrst proved by Stroock [320]. Ikeda and Watanabe [144] prove Hormander’s ¨ theorem using the following estimate for the tail of the variance of the Brownian motion: P 0

1

Wt − 0

1

√ 1 Ws ds)2 dt < ≤ 2 exp(− 7 ). 2

In [186] Kusuoka and Stroock derive Gaussian exponential bounds for the density pt (x0 , ·) of the diﬀusion Xt (x0 ) starting at x0 under hypoellipticity conditions. In [166] Kohatsu-Higa introduced in the notion of uniformly elliptic random vector and obtained Gaussian lower bound estimates for the density of a such a vector. The results are applied to the solution to the stochastic heat equation. Further applications to the potential theory for two-parameter diﬀusions are given in [76]. Malliavin calculus can be applied to study the asymptotic behavior of the fundamental solution to the heat equation (see Watanabe [344], Ben Arous, Leandre ´ [26], [27]). More generally, it can be used to analyze the asymptotic behavior of the solution stochastic partial diﬀerential equations like the stochastic heat equation (see [167]) and stochastic diﬀerential equations with two parameters (see [168]).

2.4 Stochastic partial diﬀerential equations

165

On the other hand, the stochastic calculus of variations can be used to show hypoellipticity (existence of a smooth density) under conditions that are strictly weaker than Hormander’s ¨ hypothesis. For instance, in [24] the authors allow the Lie algebra condition to fail exponentially fast on a submanifold of Rm of dimension less than m (see also [106]). In addition to the case of a diﬀusion process, Malliavin calculus has been applied to show the existence and smoothness of densities for diﬀerent types of Wiener functionals. In most of the cases analytical methods are not available and the Malliavin calculus is a suitable approach. The following are examples of this type of application: (i) Bell and Mohammed [23] considered stochastic delay equations. The asymptotic behaviour of the density of the solution when the variance of the noise tends to zero is analized in [99]. (ii) Stochastic diﬀerential equations with coeﬃcients depending on the past of the solution have been analyzed by Kusuoka and Stroock [187] and by Hirsch [134]. (iii) The smoothness of the density in a ﬁltering problem has been discussed in Bismut and Michel [43], Chaleyat-Maurel and Michel [61], and Kusuoka and Stroock [185]. The general problem of the existence and smoothness of conditional densities has been considered by Nualart and Zakai [266]. (iv) The application of the Malliavin calculus to diﬀusion processes with boundary conditions has been developed in the works of Bismut [40] and Cattiaux [60]. (v) Existence and smoothness of the density for solutions to stochastic diﬀerential equations, including a stochastic integral with respect to a Poisson measure, have been considered by Bichteler and Jacod [36], and by Bichteler et al. [35], among others. (vi) Absolute continuity of probability laws in inﬁnite-dimensional spaces have been studied by Moulinier [232], Mazziotto and Millet [220], and Ocone [271]. (vii) Stochastic Volterra equations have been considered by Rovira and Sanz-Sol´e in [295]. Among other applications of the integration-by-parts formula on the Wiener space, not related with smoothness of probability laws, we can mention the following problems: (i) time reversal of continuous stochastic processes (see F¨¨ollmer [109], Millet et al. [229], [230]),

166

2. Regularity of probability laws

(ii) estimation of oscillatory integrals (see Ikeda and Shigekawa [143], Moulinier [233], and Malliavin [209]), (iii) approximation of local time of Brownian martingales by the normalized number of crossings of the regularized process (see Nualart and Wschebor [262]), (iv) the relationship between the independence of two random variables F and G on the Wiener space and the almost sure orthogonality of ¨ u their derivatives. This subject has been developed by Ust¨ ¨nel and Zakai [333], [334]. The Malliavin calculus leads to the development of the potential theory on the Wiener space. The notion of cp,r capacities and the associated quasisure analysis were introduced by Malliavin in [208]. One of the basic results of this theory is the regular disintegration of the Wiener measure by means of the coarea measure on submanifolds of the Wiener space with ﬁnite codimension (see Airault and Malliavin [3]). In [2] Airault studies the diﬀerential geometry of the submanifold F = c, where F is a smooth nondegenerate variable on the Wiener space. [2.4] The Malliavin calculus is a helpful tool for analyzing the regularity of probability distributions for solutions to stochastic integral equations and stochastic partial diﬀerential equations. For instance, the case of the solution {X(z), z ∈ R2+ } of two-parameter stochastic diﬀerential equations driven by the Brownian sheet, discussed in Section 2.4.1, has been studied by Nualart and Sanz [256], [257]. Similar methods can be applied to the analysis of the wave equation perturbed by a two-parameter white noise (cf. Carmona and Nualart [59], and L´ ´eandre and Russo [194]). The application of Malliavin calculus to the absolute continuity of the solution to the heat equation perturbed by a space-time white noise has been taken from Pardoux and Zhang [282]. The arguments used in the last part of the proof of Theorem 2.4.4 are due to Mueller [234]. The smoothness of the density in this example has been studied by Bally and Pardoux [19]. As an application of the Lp estimates of the density obtained by means of Malliavin calculus (of the type exhibited in Exercise 2.1.5), Bally et al. [18] prove the existence of a unique strong solution for the white noise driven heat equation (2.84) when the coeﬃcient b is measurable and locally bounded, and satisﬁes a one-sided linear growth condition, while the diﬀusion coeﬃcient σ does not vanish, has a locally Lipschitz derivative, and satisﬁes a linear growth condition. Gy¨ ongy [130] has generalized this result to the case where σ is locally Lipschitz. The smoothness of the density of the vector (u(t, x1 ), . . . , u(t, xn )), where u(t, x) is the solution of a two-dimensional non-linear stochastic wave equation driven by Gaussian noise that is white in time and correlated in the space variable, has been derived in [231]. These equations were studied by

2.4 Stochastic partial diﬀerential equations

167

Dalang and Frangos in [75]. The abolute continuity of the law and the smoothness of the density for the three-dimensional non-linear stochastic wave equation has been considered in [288] and [289], following an approach to construct a solution for these equations developed by Dalang in [77]. The smoothness of the density of the projection onto a ﬁnite-dimensional subspace of the solution at time t > 0 of the two-dimensional NavierStokes equation forced by a ﬁnite-dimensional Gaussian white noise has been established by Mattingly and Pardoux in [219] (see also [132]).

3 Anticipating stochastic calculus

As we have seen in Chapter 2, the Skorohod integral is an extension of the Itˆ o integral that allows us to integrate stochastic processes that are not necessarily adapted to the Brownian motion. The adaptability assumption is replaced by some regularity condition. It is possible to develop a stochastic calculus for the Skorohod integral which is similar in some aspects to the classical Ito ˆ calculus. In this chapter we present the fundamental facts about this stochastic calculus, and we also discuss other approaches to the problem of constructing stochastic integrals for nonadapted processes (approximation by Riemann sums, development in a basis of L2 ([0, 1]), substitution methods). The last section discusses noncausal stochastic diﬀerential equations formulated using anticipating stochastic integrals.

3.1 Approximation of stochastic integrals 1 In order to deﬁne the stochastic integral 0 ut dW Wt of a not necessarily adapted process u = {ut , t ∈ [0, 1]} with respect to the Brownian motion W , one could use the following heuristic approach. First approximate u by a sequence of step processes un , then deﬁne the stochastic integral of each process un as a ﬁnite sum of the increments of the Brownian motion multiplied by the values of the process in each interval, and ﬁnally try to check if the sequence of integrals converges in some topology. What happens is that diﬀerent approximations by step processes will produce diﬀerent types of integrals. In this section we discuss this approach, and

170

3. Anticipating stochastic calculus

in particular we study two types of approximations, one leading to the Skorohod integral, and a second one that produces a Stratonovich-type stochastic integral.

3.1.1 Stochastic integrals deﬁned by Riemann sums In this section we assume that {W (t), t ∈ [0, 1]} is a one-dimensional Brownian motion, deﬁned in the canonical probability space (Ω, F, P ). We denote by π an arbitrary partition of the interval [0, 1] of the form π = {0 = t0 < t1 < · · · < tn = 1}. We have to take limits (in probability, or in Lp (Ω), p ≥ 1) of families of random variables Sπ , depending on π, as the norm of π (deﬁned as |π| = sup0≤i≤n−1 (ti+1 −ti )) tends to zero. Notice ﬁrst that this convegence is equivalent to the convergence along any sequence of partitions whose norms tend to zero. In most of the cases it suﬃces to consider increasing sequences, as the next technical lemma explains. Lemma 3.1.1 Let Sπ be a family of elements of some complete metric space (V, d) indexed by the class of all partitions of [0, 1]. Suppose that for any ﬁxed partition π 0 we have lim d(S Sπ∨π0 , Sπ ) = 0,

|π|→0

(3.1)

where π ∨ π 0 denotes the partition induced by the union of π and π 0 . Then the family Sπ converges to some element S if and only if for any increasing sequence of partitions {π(k), k ≥ 1} of [0, 1], such that |π(k)| → 0, the sequence Sπ(k) converges to S as k tends to inﬁnity. Proof: Clearly, the convergence of the family Sπ implies the convergence of any sequence Sπ(k) with |π(k)| → 0 to the same limit. Conversely, suppose (k) with | π (k)| → 0, but that Sπ (k) → S for any increasing sequence π there exists an > 0 and a sequence π(k) with |π(k)| → 0 such that d(S Sπ(k) , S) > for all k. Then we ﬁx k0 and by (3.1) we can ﬁnd a k1 such that k1 > k0 and d(S Sπ(k0 )∨π(k1 ) , Sπ(k1 ) ) < . 2 Next we choose k2 > k1 large enough so that d(S Sπ(k0 )∨π(k1 )∨π(k2 ) , Sπ(k2 ) ) < , 2 and we continue recursively. Set π (n) = π(k0 ) ∨ π(k1 ) ∨ · · · ∨ π(kn ). Then after the nth step we have Sπ(kn ) , S) − d(S Sπ (n) , Sπ(kn ) ) > . d(S Sπ (n) , S) ≥ d(S 2 Then π (n) is an increasing sequence of partitions such that the sequence of norms | π (n)| tends to zero but d(S Sπ (n) , S) > 2 , which completes the proof by contradiction.

3.1 Approximation of stochastic integrals

171

1 Consider a measurable process u = {ut , t ∈ [0, 1]} such that 0 |ut |dt < ∞ a.s. For any partition π we introduce the following step process: uπ (t) =

n−1 i=0

If E

1 0

1 ti+1 − ti

ti+1

ti

us ds 1(ti ,ti+1 ] (t).

(3.2)

|ut |dt < ∞ we deﬁne the step process

u π (t) =

n−1 i=0

1 ti+1 − ti

ti+1

ti

E(us |F F[ti ,ti+1 ]c )ds 1(ti ,ti+1 ] (t).

(3.3)

We recall that F[cti ,ti+1 ] denotes the σ-ﬁeld generated by the increments Wt − Ws , where the interval (s, t] is disjoint with [ti , ti+1 ]. The next lemma presents in which topology the step processes uπ and π u are approximations of the process u. Lemma 3.1.2 Suppose that u belongs to L2 ([0, 1]×Ω). Then, the processes π converge to the process u in the norm of the space L2 ([0, 1] × Ω) uπ and u as |π| tends to zero. Furthermore, these convergences also hold in L1,2 whenever u ∈ L1,2 . Proof: The convergence uπ → u in L2 ([0, 1] × Ω) as |π| tends to zero can be proved as in Lemma 1.1.3, but for the convergence of u π we need a diﬀerent argument. π satisfy condition (3.1) with One can show that the families uπ and u 2 V = L ([0, 1] × Ω) (see Exercise 3.1.1). Consequently, by Lemma 3.1.1 it suﬃces to show the convergence along any ﬁxed increasing sequence of partitions π(k) such that |π(k)| tends to zero. In the case of the family uπ , we can regard uπ as the conditional expectation of the variable u, in the probability space [0, 1] × Ω, given the product σ-ﬁeld of the ﬁnite algebra of parts of [0, 1] generated by π times F. Then the convergence of uπ to u in L2 ([0, 1] × Ω) along a ﬁxed increasing sequence of partitions follows from the martingale convergence theorem. For the family u π the argument of the proof is as follows. Let π(k) be an increasing sequence of partitions such that |π(k)| → 0. Set π(k) = {0 = tk0 < tk1 < · · · < tknk = 1}. For any k we consider the σ-ﬁeld G k of parts of [0, 1] × Ω generated by the sets (tki , tki+1 ] × F , where k 0 ≤ i ≤ nk − 1 and F ∈ F[tki ,tki+1 ]c . Then notice that u π(k) = E(u|G ), where E denotes the mathematical expectation in the probability space [0, 1] × Ω. By the martingale convergence theorem, u π(k) converges to some 2 element u in L ([0, 1] × Ω). We want to show that u = u. The diﬀerence v = u − u is orthogonal to L2 ([0, 1] × Ω, Gk ) for every k. Consequently, for any ﬁxed k ≥ 1, such a process v satisﬁes I ×F v(t, ω)dtdP = 0 for any F ∈ F[tki ,tki+1 ]c and for any interval I ⊂ [tki , tki+1 ] in π(m) with m ≥ k. Therefore,

172

3. Anticipating stochastic calculus

E(v(t)|F F[tki ,tki+1 ]c ) = 0 for all (t, ω) almost everywhere in [tki , tki+1 ] × Ω. Therefore, for almost all t, with respect to the Lebesgue measure, the above conditional expectation is zero for any i, k such that t ∈ [tki , tki+1 ]. This implies that v(t, ω) = 0 a.s., for almost all t, and the proof of the ﬁrst part of the lemma is complete. In order to show the convergence in L1,2 we ﬁrst compute the derivatives π using Proposition 1.2.8: of the processes uπ and u π

Dr u (t) =

n−1

1 ti+1 − ti

i=0

ti+1

Dr us ds 1(ti ,ti+1 ] (t),

ti

and π (t) Dr u

=

n−1

1 ti+1 − ti

i=0

ti+1

ti

E(Dr us |F F[ti ,ti+1 ]c )ds

×1(ti ,ti+1 ] (t)1(ti ,ti+1 ]c (r). Then, the same arguments as in the ﬁrst part of the proof will give the desired convergence. Now consider the Riemann sums associated to the preceding approximations: ti+1 n−1 1 us ds (W (ti+1 ) − W (ti )) Sπ = t − ti ti i=0 i+1 and S π =

n−1 i=0

1 ti+1 − ti

ti+1

ti

E(us |F F[ti ,ti+1 ]c )ds (W (ti+1 ) − W (ti )).

Notice that from Lemma 1.3.2 the processes u π are Skorohod integrable 2 for any process u in L ([0, 1] × Ω) and that S π = δ( uπ ). On the other hand, for the process uπ to be Skorohod integrable we need some additional conditions. For instance, if u ∈ L1,2 , then uπ ∈ L1,2 ⊂ Dom δ, and we have δ(uπ ) = S π −

n−1 i=0

1 ti+1 − ti

ti+1

ti+1

Ds ut dsdt. ti

(3.4)

ti

In conclusion, from Lemma 3.1.2 we deduce the following results: (i) Let u ∈ L2 ([0, 1] × Ω). If the family S π converges in L2 (Ω) to some limit, then u is Skorohod integrable and this limit is equal to δ(u).

3.1 Approximation of stochastic integrals

173

(ii) Let u ∈ L1,2 . Then both families S π = δ( uπ ) and δ(uπ ) converge in 2 L (Ω) to δ(u). Let us now discuss the convergence of the family S π . Notice that π

S =

1

ut Wtπ dt,

0

where Wtπ =

n−1 i=0

W (ti+1 ) − W (ti ) 1(ti ,ti+1 ] (t). ti+1 − ti

(3.5)

Deﬁnition 3.1.1 We say that a measurable process u = {ut , 0 ≤ t ≤ 1} 1 such that 0 |ut |dt < ∞ a.s. is Stratonovich integrable if the family S π converges in probability as |π| → 0, and in this case the limit will be denoted 1 Wt . by 0 ut ◦ dW From (3.4) we see that for a given process u to be Stratonovich integrable it is not suﬃcient that u ∈ L1,2 . In fact, the second summand in (3.4) can be regarded as an approximation of the trace of the kernel Ds ut in [0, 1]2 , and this trace is not well deﬁned for an arbitrary square integrable kernel. Let us introduce the following deﬁnitions: Let X ∈ L1,2 and 1 ≤ p ≤ 2. We denote by D+ X (resp. D− X) the element of Lp ([0, 1] × Ω) satisfying

1

lim

(resp. n→∞

(3.6)

E(|Ds Xt − (D− X)s |p )ds = 0).

(3.7)

1 0 s
lim

E(|Ds Xt − (D+ X)s |p )ds = 0

sup

n→∞

1

sup 1 0 (s− n )∨0≤t<s

1,2 1,2 We denote by Lp+ (resp. Lp− ) the class of processes in L1,2 such that (3.6) 1,2 1,2 1,2 (resp. (3.7)) holds. We set Lp = Lp+ ∩ Lp− . For X ∈ Lp1,2 we write

(∇X)t = (D+ X)t + (D− X)t .

(3.8)

Ds Xt is continuous Let X ∈ L1,2 . Suppose that the mapping (s, t) → from a neighborhood of the diagonal Vε = {|s − t| < ε} into Lp (Ω). Then X ∈ Lp1,2 and (D+ X)t = (D− X)t = Dt Xt . The following proposition provides an example of a process in the class L21,2 .

174

3. Anticipating stochastic calculus

Proposition 3.1.1 Consider a process of the form

t

X t = X0 +

t

us dW Ws + 0

vs ds,

(3.9)

0

where X0 ∈ D1,2 and u ∈ D2,2 (H), and v ∈ L1,2 . Then, X belongs to the class L21,2 and t t ut + Dt X0 + Dt vr dr + Dt ur dW Wr , 0 0 t t = Dt X0 + Dt vr dr + Dt ur dW Wr .

(D+ X)t

=

(D− X)t

0

Proof:

(3.10) (3.11)

0

First notice that X belongs to L1,2 , and for any s we have Ds Xt = us 1[0,t] (s) + Ds X0 +

t

Ds vr dr + 0

t

Ds ur dW Wr . 0

Thus,

1

sup E(|Ds Xt − (D− X)s |2 )ds )∨0
1 s− n

and this converges to zero as n tends to inﬁnity. In a similar way we show that (D+ X)t exists and is given by (3.10). 1,2,f In a similar way, Lp− is the class of processes X in L1,2,f such that there − exists an element D X ∈ Lp ([0, 1] × Ω) for which (3.7) holds. Suppose that Xt is given by (3.9), where X0 ∈ D1,2 , u ∈ LF and v ∈ L1,2,f , then, X 1,2,f belongs to the class L2− and (D− X)t is given by (3.11).

Then we have the following result, which gives suﬃcient conditions for the existence of the Stratonovich integral and provides the relation between the Skorohod and the Stratonovich integrals. 1,2 . Then u is Stratonovich integrable and Theorem 3.1.1 Let u ∈ L1,loc

0

1

ut ◦ dW Wt = 0

1

1 ut dW Wt + 2

1

(∇u)t dt. 0

(3.12)

3.1 Approximation of stochastic integrals

175

Proof: By the usual localization argument we can assume that u ∈ L11,2 . Then, from Eq. (3.4) and the above approximation results on the Skorohod integral, it suﬃces to show that n−1 i=0

1 ti+1 − ti

ti+1

ti

ti+1

ti

Dt us dsdt →

1 2

1

(∇u)s ds, 0

in probability, as |π| → 0. We will show that the expectation n−1 ti+1 ti+1 1 1 1 + D u t dt dt (Dt us )ds − E ti+1 − ti ti 2 0 t i=0

converges to zero as |π| → 0. A similar result can be proved for the operator D− , and the desired convergence would follow. We majorize the above expectation by the sum of the following two terms: n−1 ti+1 ti+1 + 1 dt (Dt us − D u t )ds E t − t i ti t i=0 i+1 ti+1 n−1 ti+1 − t 1 1 + + +E D u t dt − D u t dt ti t − ti 2 0 i=0 i+1 1 ≤ sup E(|Dt us − D+ u t |)dt 0 t≤s≤(t+|π|)∧1

1 ti+1 − t n−1 1 + D u t dt . 1(ti ,ti+1 ] (t) − +E 0 t − t 2 i+1 i i=0 The ﬁrst term in the above expression tends to zero by the deﬁnition of the class L11,2 . For the second term we will use the convergence of the

n−1 i+1 −t 1 functions i=0 tti+1 −ti 1(ti ,ti+1 ] (t) to the constant 2 in the weak topology 2 of L ([0, 1]). This weak convergence implies that 1 ti+1 − t + n−1 1 D u t dt 1(t ,t ] (t) − t − ti i i+1 2 0 i=0 i+1 converges a.s. to zero as |π| → 0. Finally, the convergence in L1 (Ω) follows by dominated convergence, using the deﬁnition of the space L11,2 . Remarks: 1. If the mapping (s, t) → Ds ut is continuous from Vε = {|s − t| < ε} into 1 1 L (Ω), then the second summand in formula (3.12) reduces to 0 Dt ut dt. 2. Suppose that X is a continuous semimartingale of the form (1.23). Then the Stratonovich integral of X exists on any interval [0, t] and coincides

176

3. Anticipating stochastic calculus

with the limit in probability of the sums (1.24). That is, we have

t

Xs ◦ dW Ws = 0

0

t

1 Xs dW Ws + X, W t , 2

where X, W denotes the joint quadratic variation of the semimartingale X and the Brownian motion. Suppose in addition that X ∈ L11,2 . In that case, we have (D− X)t = 0 (because X is adapted), and consequently,

1

D+ X

0

dt = X, W 1 = lim t

|π|↓0

n−1

(X(ti+1 ) − X(ti ))(W (ti+1 ) − W (ti )),

i=0

where π denotes a partition of [0, 1]. In general, for processes u ∈ L11,2 that are continuous in L2 (Ω), the joint quadratic variation of the process u and the Brownian motion coincides with the integral

1

0

( D+ u t − D− u t )dt

(see Exercise 3.1.2). Thus, the joint quadratic variation does not coincide in general with the diﬀerence between the Skorohod and Stratonovich integrals. The approach we have described admits diverse extensions. For instance, we can use approximations of the following type: n−1

((1 − α)u(ti ) + αu(ti+1 ))(W (ti+1 ) − W (ti )),

(3.13)

i=0

where α is a ﬁxed number between 0 and 1. Assuming that u ∈ L11,2 and E(ut ) is continuous in t, we can show (see Exercise 3.1.3) that expression (3.13) converges in L1 (Ω) to the following quantity: δ(u) + α 0

1

+ D u t dt + (1 − α)

1 0

− D u t dt.

(3.14)

For α = 12 we obtain the Stratonovich integral, and for α = 0 expression (3.14) is the generalization of the Itˆ oˆ integral studied in [14, 30, 298].

3.1.2

The approach based on the L2 development of the process

Suppose that {W (A), A ∈ B, µ(A) < ∞} is an L2 (Ω)-valued Gaussian measure associated with a measure space (T, B, µ). We ﬁx a complete orthonormal system {ei , i ≥ 1} in the Hilbert space H = L2 (T ). We can

3.1 Approximation of stochastic integrals

177

compute the random Fourier coeﬃcients of the paths of the process u in that basis: ∞ u(t) = u, ei H ei (t). i=1

Then one can deﬁne the stochastic integral of u (wihch we will denote by 1 u ∗ dW Wt , following Rosinski [293]) as the sum of the series 0 t ut ∗ dW Wt = T

∞

u, ei H W (ei ),

(3.15)

i=1

provided that it converges in probability (or in L2 (Ω)) and the sum does not depend on the complete orthonormal system we have chosen. We will call it the L2 -integral. We remark that if u ∈ L1,2 , then using (1.48) we have for any i u, ei H δ(ei ) = δ(ei u, ei H ) + Ds ut ei (s)ei (t)µ(ds)µ(dt). T

T

Consequently, if u ∈ L and the kernel Ds ut has a summable trace for Wt exists, and we have the following all ω a.s., then the integral T ut ∗ dW relation (cf. Nualart and Zakai [263, Proposition 6.1]): ut ∗ dW Wt = ut dW Wt + T(Du). (3.16) 1,2

T

T

The following result (cf. Nualart and Zakai [267]) establishes the relationship between the Stratonovich integral and the L2 -integral when T = [0, 1] and µ is the Lebesgue measure. 1 Theorem 3.1.2 Let u be a measurable process such that 0 u2t dt < ∞ a.s. 1 Wt exists, u is Stratonovich integrable, and Then if the integral 0 ut ∗ dW both integrals coincide. Proof: It suﬃces to show that for any increasing sequence of partitions {π(n), n ≥ 1} whose norm tends to zero and π(n+1) is obtained by reﬁning 1 Wt as n tends to π(n) at one point, the sequence S π(n) converges to 0 ut ∗dW inﬁnity. The idea of the proof is to produce a particular complete orthonormal system for which S π(n) will be the partial sum of series (3.15). Without any loss of generality we may assume that π(1) = {0, 1}. Set e1 = 1. For n ≥ 1 deﬁne en+1 as follows. By our assumption π(n+1) reﬁnes π(n) in one point only. Assume that the point of reﬁnement τ belongs to some interval (sn , tn ) of the partition π(n). Then we set en+1 (t) =

tn − τ (τ − sn )(tn − sn )

12

178

3. Anticipating stochastic calculus

if t ∈ (sn , τ ],

en+1 (t) = −

τ − sn (tn − τ )(tn − sn )

12

if t ∈ (τ , tn ], and en+1 (t) = 0 for t ∈ (sn , tn ]c . In this form we construct a complete orthonormal system in L2 ([0, 1]), which can be considered as a modiﬁed Haar system. Finally, we will show by induction that the partial sums of series (3.15) coincide with the approximations of the Stratonovich integral corresponding to the sequence of partitions π(n). First notice that S π(n+1) diﬀers from S π(n) only because of changes taking place in the interval (sn , tn ]. Therefore, tn 1 π(n+1) π(n) −S =− ut dt (W (tn ) − W (sn )) S tn − sn sn τ 1 + ut dt (W (τ ) − W (sn )) τ − sn sn tn 1 + ut dt (W (tn ) − W (τ )). (3.17) tn − τ τ On the other hand, 1 1 en+1 (t)ut dt en+1 (t)dW Wt 0 0 & & '1 '1 τ − sn 2 tn tn − τ 2 τ 1 = ut dt − ut dt tn − sn τ − sn tn − τ sn τ & & '1 '1 tn − τ 2 τ − sn 2 (W (τ ) − W (sn )) − (W (tn ) − W (τ )) , × τ − sn tn − τ which is equal to 1 tn − τ τ ut dt (W (τ ) − W (sn )) tn − sn τ − sn sn τ tn −( ut dt)(W (tn ) − W (τ ))) − ut dt (W (τ ) − W (sn )) sn

τ − sn + tn − τ

tn

τ

ut dt (W (tn ) − (W (τ )) .

(3.18)

τ

Comparing (3.17) with (3.18) obtains the desired result. Using the previous theorem, we can show the existence of the Stratonovich integral under conditions that are diﬀerent from those of Theorem 3.1.1. 1,2 Proposition 3.1.2 Let u be a process in Lloc . Suppose that for all ω a.s. the integral operator from H into H associated with the kernel Du(ω) is a

3.1 Approximation of stochastic integrals

179

nuclear (or a trace class) operator. Then u is Stratonovich integrable, and we have 1 ut ◦ dW Wt = δ(u) + T(Du). (3.19) 0

Proof: We have seen that for any complete orthonormal system {ei , i ≥ 1} in H the series 1 1 ∞ ut ei (t)dt ei (s)dW Ws i=1

0

0

converges in probability to δ(u) + T(Du). Then Proposition 3.1.2 follows from Theorem 3.1.2.

Exercises 3.1.1 Let u ∈ L2 ([0, 1] × Ω). Show that the families uπ and u ˆπ deﬁned 2 in Eqs. (3.2) and (3.3) satisfy condition (3.1) with V = L ([0, 1] × Ω). If u ∈ L1,2 , then (3.1) holds with V = L1,2 . 3.1.2 Let u ∈ L11,2 be a process continuous in L2 (Ω). Show that n−1

(u(ti+1 ) − u(ti ))(W (ti+1 ) − W (ti ))

i=0

1 converges in L1 (Ω) to 0 ((D+ u)t − (D− u)t )dt as |π| tends to zero (see Nualart and Pardoux [249, Theorem 7.6]). 3.1.3 Show the convergence of (3.13) to (3.14) in L1 (Ω) as |π| tends to zero. The proof is similar to that of Theorem 3.1.1. 3.1.4 Show that the process ut = W1−t is not L2 -integrable but that it is Stratonovich integrable and 1 1 W1−t ◦ dW Wt = W 12 + 2 W1−t dW Wt . 2

0

0

functions Hint: Consider the sequences √ of {φn } and {ψ n } of orthonormal √ in L2 ([0, 1]) given by φn (t) = 2 cos(2πnt) and ψ n (t) = 2 sin(2πn(1−t)), n ≥ 1, t ∈ [0, 1] (see Rosinski [293]). 3.1.5 Let ψ be a nonnegative C ∞ function on [−1, 1] whose integral is 1 and such that ψ(x) = ψ(−x). Consider the approximation of the identity ψ (x) = 1 ψ( x ). Fix a process u ∈ L1,2 , and deﬁne I (u) =

1

ut (ψ ∗ W )t dt.

0 1

Show that I (u) converges in L (Ω) to the Stratonovich integral of u. The 1,2 . convergence holds in probability if u ∈ Lloc

180

3. Anticipating stochastic calculus

3.1.6 Let u = {ut , t ∈ [0, 1]} be a Stratonovich integrable process and let F be a random variable. Show that F ut is Stratonovich integrable and that

1

F ut ◦ dW Wt = F

0

1

ut ◦ dW Wt .

0

3.2 Stochastic calculus for anticipating integrals In this section we will study the properties of the indeﬁnite Skorohod integral as a stochastic process. In particular we discuss the regularity properties of its paths (continuity and quadratic variation) and obtain a version of the Itoˆ formula for the indeﬁnite Skorohod integral.

3.2.1

Skorohod integral processes

Fix a Brownian motion W = {W (t), t ∈ [0, 1]} deﬁned on the canonical probability space (Ω, F, P ). Suppose that u = {u(t), 0 ≤ t ≤ 1} is a Skorohod integrable process. In general, a process of the form u1(s,t] may not be Skorohod integrable (see Exercise 3.2.1). Let us denote by Ls the set of processes u such that u1[0,t] is Skorohod integrable for any t ∈ [0, 1]. Notice that the space L1,2 is included into Ls . Suppose that u belongs to Ls , and deﬁne t

X(t) = δ(u1[0,t] ) =

us dW Ws .

(3.20)

0

The process X is not adapted, and its increments satisfy the following orthogonality property: Lemma 3.2.1 For any process u ∈ Ls we have t E( ur dW Wr |F F[s,t]c ) = 0

(3.21)

s

for all s < t, where, as usual, F[s,t]c denotes the σ-ﬁeld generated by the increments of the Brownian motion in the complement of the interval [s, t]. Proof: To show (3.21) it suﬃces to take an arbitrary F[s,t]c -measurable random variable F belonging to the space D1,2 , and check that E(F s

t

ur dW Wr ) = E( 0

1

ur Dr F 1[s,t] (r)dr) = 0,

which holds due to the duality relation (1.42) and Corollary 1.2.1. t Let us denote by M the class of processes of the form X(t) = 0 us dW Ws , where u ∈ Ls . One can show (see [237]) that if a process X = {Xt , t ∈ [0, 1]}

3.2 Stochastic calculus for anticipating integrals

181

satisﬁes X(0) = 0, E(X(t)2 ) < ∞, for all t, and sup

n−1

E[(X(tj+1 ) − X(tj ))2 ] < ∞,

π={0=t0
then X belongs to M if and only if condition (3.21) is satisﬁed. The necessity of (3.21) has been proved in Lemma 3.2.1. A process X in M is continuous in L2 . However, there exist processes u t Ws does not have a continuous in Ls such that the indeﬁnite integral 0 us dW version (see Exercise 3.2.3). This can be explained by the fact that the process X is not a martingale, and we do not dispose of maximal inequalities to show the existence of a continuous version.

3.2.2 Continuity and quadratic variation of the Skorohod integral If u belongs to L1,2 , then, under some integrability assumptions, we will show the existence of a continuous version for the indeﬁnite Skorohod integral of u by means of Kolmogorov’s continuity criterion. To do this we need the following Lp (Ω) estimates for the Skorohod integral that are deduced by duality from Meyer’s inequalities (see Proposition 1.5.8): Proposition 3.2.1 Let u be a stochastic process in L1,2 , and let p > 1. Then we have 1 1 1 1 1 2 2 2 2 (Ds ut ) dsdt) p . (3.22) δ(u)p ≤ Cp ( (E(ut )) dt) + ( 0

0

0

The following proposition provides a suﬃcient condition for the existence of a continuous version of the indeﬁnite Skorohod integral. Proposition 3.2.2 Let a process in the class L1,2 . Suppose that for 1u be p 1 some p > 2 we have E 0 ( 0 (Ds ut )2 ds) 2 dt < ∞. Then the integral process t { 0 us dW Ws , 0 ≤ t ≤ 1} has a continuous version. Proof: We may assume that E(ut ) = 0 for any t, because the Gaussian t t Ws always has a continuous version. Set Xt = 0 us dW Ws . process 0 E(us )dW Applying Proposition 3.2.1 and H¨ o¨lder’s inequality, we obtain for s < t t E(|Xt − Xs |p ) ≤ Cp E(|

1

p

(Dθ ur )2 dθdr| 2 ) s 0 p2 t 1 p −1 2 (Dθ ur ) dθ dr . ≤ Cp |t − s| 2 E s

0

182

3. Anticipating stochastic calculus

p2 1 Set Ar = 0 (Dθ ur )2 dθ . Fix an exponent 2 < α < 1 + p2 , and assume that p is close to 2. Applying Fubini’s theorem we can write 1 1 |Xt − Xs |p dsdt E |t − s|α 0 0 p

≤ 2C Cp E =

2C Cp α − p2

{s
{s

|t − s| 2

t

−1−α

Ar drdsdt s

p

|r − s| 2

2C C p = α − p2 p2 + 1 − α

−α

1

p

− |1 − s| 2

−α

E(Ar )drds p

p

r 2 + 1 − α + 1 − |1 − r| 2

+1−α

E(Ar )dr < ∞.

0

Hence, the random variable deﬁned by 1 1 |Xt − Xs |p dsdt Γ= |t − s|α 0 0 satisﬁes E(Γ) < ∞ and by the results of Section A.3 we obtain 1

|Xt − Xs | ≤ cp,α Γ p |t − s|

α−2 p

for some constant cp,α . k,p For every p > 1 and any positive integer k we will denote by L the space Lp ([0, 1]; Dk,p ) ⊂ Dk,p (L2 [0, 1]). Notice that for p = 2 and k = 1 this deﬁnition is consistent with the previous notation for the space L1,2 . The above proposition implies that for a process u in L1,p loc , with p > 2, t Ws has a continuous version. Furthermore, if the Skorohod integral 0 us dW u in L1,p , p > 2, we have t p us dW Ws < ∞. E sup t∈[0,1]

0

It is possible to show the existence of a continuous version under diﬀerent hypotheses (see Exercises 3.2.4, and 3.2.5.). The next result will show the existence of a nonzero quadratic variation for the indeﬁnite Skorohod integral (see [249]). 1,2 . Then Theorem 3.2.1 Suppose that u is a process of the space Lloc

ti+1 n−1 i=0

ti

2 us dW Ws

→

1

u2s ds,

(3.23)

0

in probability, as |π| → 0, where π runs over all ﬁnite partitions {0 = t0 < t1 < · · · < tn = 1} of [0, 1]. Moreover, the convergence is in L1 (Ω) if u belongs to L1,2 .

3.2 Stochastic calculus for anticipating integrals

183

Proof: We will describe the details of the proof only for the case u ∈ L1,2 . The general case would be deduced by an easy argument of localization. For any process u in L1,2 and for any partition π = {0 = t0 < t1 < · · · < tn = 1} we deﬁne V π (u) =

ti+1 n−1

2 us dW Ws

.

ti

i=0

Suppose that u and v are two processes in L1,2 . Then we have E (|V π (u) − V π (v)|)

≤

ti+1 n−1

E

ti+1 n−1

2 12 (us + vs )dW Ws

ti

i=0

≤

(us − vs )dW Ws

ti

i=0

× E

2 12

u − vL1,2 u + vL1,2 .

(3.24)

It follows from this estimate that it suﬃces to show the result for a class of processes u that is dense in L1,2 . So we can assume that ut =

m−1

Fj 1(sj ,sj+1 ] ,

j=0

where Fj is a smooth random variable for each j, and 0 = s0 < · · · < sm = 1. We can assume that the partition π contains the points {s0 , . . . , sm }. In this case we have 2 ti+1 m−1 π Ds Fj ds Fj (W (ti+1 ) − W (ti )) − V (u) = ti

j=0 {i:sj ≤ti <sj+1 }

=

m−1 j=0

⎡ ⎣

Fj2 (W (ti+1 ) − W (ti ))2

{i:sj ≤ti <sj+1 }

−2(W (ti+1 ) − W (ti ))

ti+1

Ds Fj ds + ti

2 )

ti+1

Ds Fj ds

.

ti

With the properties of the quadratic variation of the Brownian motion, this converges in L1 (Ω) to m−1 j=0

as |π| tends to zero.

Fj2 (sj+1 − sj ) =

1

u2s ds,

0

184

3. Anticipating stochastic calculus

1,2 As a consequence of the previous result, if u ∈ Lloc is a process such that t the Skorohod integral 0 us dW Ws has a continuous version with bounded variation paths, then u = 0. Theorem 3.2.1 also holds if we assume that u belongs to LF loc .

3.2.3 Ito’s ˆ formula for the Skorohod and Stratonovich integrals In this section we will show the change-of-variables formula for the indefinite Skorohod integral. We start with the following version of this formula. 4 2,2 the space of processes u ∈ L2,2 such that E(DuL2 ([0,1]2 ) ) < Denote by L(4) ∞. Theorem 3.2.2 Consider a process of the form t t us dW Ws + vs ds, Xt = X0 + 0

(3.25)

0

1,2 2,2 1,2 , u ∈ L(4),loc and v ∈ Lloc . Suppose that the process X where X0 ∈ Dloc has continuous paths. Let F : R → R be a twice continuously diﬀerentiable 1,2 and we have function. Then F (Xt )ut belongs to Lloc

F (Xt )

t 1 t = F (X0 ) + F (Xs )dXs + F (Xs )u2s 2 0 0 t + F (Xs )(D− X)s us ds.

(3.26)

0

Proof: Suppose that (Ωn,1 , X0n ), (Ωn,2 , un ) and (Ωn,3 , v n ) are localizing sequences for X0 , u and v, respectively. For each positive integer k let ψ k be a smooth function such that 0 ≤ ψ k ≤ 1, ψ k (x) = 0 if |x| ≥ k + 1, and ψ k (x) = 1 if |x| ≤ k. Deﬁne un,k t

=

unt ψ k

1

(uns )2 ds

.

(3.27)

0

Set Xtn,k = X0n +

t 0

un,k Ws + s dW

t 0

vsn ds and consider the familuy of sets

n,k

G

n,1

=Ω

∩Ω

n,2

∩Ω

n,3

∩

sup |Xt | ≤ k

0≤t≤1

∩

1

(uns )2 ds

≤k .

0

Deﬁne F k = F ψ k . Then, by a localization argument, it suﬃces to show the result for the processes X0n , un,k , and v n and for the function F k . In this 1 2,2 , v ∈ L1,2 , 0 u2s ds ≤ k, and way we can assume that X0 ∈ D1,2 , u ∈ L(4) that the functions F , F and F are bounded.

3.2 Stochastic calculus for anticipating integrals

185

Set tni = 2itn , 0 ≤ i ≤ 2n . As usual, the basic argument in proving a change-of-variables formula is Taylor development. Going up to the second order, we get F (Xt )

= F (X0 ) +

n 2 −1

F (X(tni ))(X(tni+1 ) − X(tni ))

i=0 n 2 −1

+

i=0

1 F (X i )(X(tni+1 ) − X(tni ))2 , 2

where X i denotes a random intermediate point between X(tni ) and X(tni+1 ). Now the proof will be decomposed in several steps. Step 1. Let us show that n 2 −1

F

(X i )(X(tni+1 )

−

X(tni ))2

t

→

F (Xs )u2s ds,

(3.28)

0

i=0

in L1 (Ω), as n tends to inﬁnity. The increment (X(tni+1 ) − X(tni ))2 can be decomposed into

2

tn i+1

us dW Ws

+

tn i

2

tn i+1

vs ds

+2

tn i

tn i+1

tn i+1

us dW Ws tn i

vs ds .

tn i

The contribution of the last two terms to the limit (3.28) is zero. In fact, we have n n 2 2 −1 t ti+1 −n E F (X i ) vs ds ≤ F ∞ t2 E(vs2 )ds, tn 0 i=0 i and

n n n −1 2 ti+1 ti+1 F (X i ) us dW Ws vs ds E n n ti ti i=0

≤

−n

F ∞ t2

t

E(vs2 )ds 0

12

uL1,2 .

Therefore, it suﬃces to show that n 2 n t 2 −1 ti+1 F (X i ) us dW Ws → F (Xs )u2s ds i=0

tn i

0

in L1 (Ω), as n tends to inﬁnity. Suppose that n ≥ m, and for any i = (m) 1, . . . , n let us denote by ti the point of the mth partition that is closer

186

3. Anticipating stochastic calculus

to tni from the left. Then we have n n 2 2 ti+1 t −1 2 F (X ) u dW W − F (X )u ds i s s s s tn 0 i=0 i n 2 2 tni+1 −1 (m) ≤ [F (X i ) − F (X(ti ))] us dW Ws tn i=0 i m ⎡ ⎤ 2 n 2 tni+1 ti+1 −1 2 ⎦ ⎣ + F (X(tm )) u dW W − u ds s s j s tn tn j=0 m m i i i:tn i ∈[tj ,tj+1 ) m 2 −1 tm t j+1 2 2 + F (X(tm )) u ds − F (X )u ds s s j s tm 0 j=0 j = b1 + b 2 + b 3 . The expectation of the term b3 can be bounded by kE

sup |s−r|≤t2−m

|F (Xs ) − F (Xr )| ,

which converges to zero as m tends to inﬁnity by the continuity of the process Xt . In the same way the expectation of b1 is bounded by ⎛ 2 ⎞ n 2 −1 tn i+1 (3.29) |F (Xs ) − F (Xr )| us dW Ws ⎠ . E ⎝ sup |s−r|≤t2−m

tn i

i=0

Letting ﬁrst n tends to inﬁnity and applying Theorem 3.2.1, (3.29) converges to 1 2 E sup |F (Xs ) − F (Xr )| us ds , |s−r|≤t2−m

0

which tends to zero as m tends to inﬁnity. Finally, the term b2 converges to zero in L1 (Ω) as n tends to inﬁnity, for any ﬁxed m, due to Theorem 3.2.1. Step 2. Clearly, n 2 −1

F

(X(tni )

i=0

tn i+1

vs ds tn i

→

t

F (Xs )vs ds

(3.30)

0

in L1 (Ω) as n tends to inﬁnity. Step 3. From Proposition 1.3.5 we deduce tni+1 tni+1 n n F (X(ti )) us dW Ws = F (X(ti ))us dW Ws + tn i

tn i

tn i+1

tn i

Ds [F (X(tni ))]us ds.

3.2 Stochastic calculus for anticipating integrals

187

Therefore, we obtain n 2 −1

i=0

+

F

(X(tni ))

n 2 −1 tn i+1

tn i

i=0

tn i+1

us dW Ws = tn i

n 2 −1 tn i+1

tn i

i=0

F (X(tni ))us dW Ws

F (X(tni ))Ds X(tni )us ds.

(3.31)

Let us ﬁrst show that n 2 −1 tn i+1

i=0

tn i

F (X(tni ))Ds X(tni )us ds →

t

F (Xs )(D− X)s us ds

(3.32)

0

in L1 (Ω) as n tends to inﬁnity. We have n n t −1 2 ti+1 n n − E F (X(ti ))Ds X(ti )us ds − F (Xs )(D X)s us ds n 0 i=0 ti n tni+1 −1 2 ! " ≤ E Ds X(tni ) − (D− X)s us ds F (X(tni )) n t i i=0 2n −1 n ti+1 +E [F (X(tni )) − F (Xs )] (D− X)s us ds n t i i=0 1/2 t ≤ F ∞ E u2s ds 0

2 ×

t

sup 0 s−t2−n ≤r≤s

2 E Ds Xr − (D− X)s ds

+E

sup |s−r|≤t2−n

:

=

dn1

+

|F (Xs ) − F (Xr )|

t

31/2

− (D X)s us ds

0

dn2 .

t The term dn2 tends to zero as n tends to inﬁnity because E 0 |(D− X)s us | ds < ∞. The term dn1 tends to zero as n tends to inﬁnity because X belongs to L21,2 by Proposition 3.1.1. As a consequence of the convergences (3.28), (3.30) and (3.32) we have proved that the sequence

An :=

n 2 −1 tn i+1

i=0

tn i

F (X(tni ))us dW Ws

188

3. Anticipating stochastic calculus

converges in L1 (Ω) as n tends to inﬁnity to t 1 t Φt : = F (Xt ) − F (X0 ) − F (Xs )vs ds − F (Xs )u2s 2 0 0 t − F (Xs )(D− X)s us ds. (3.33) 0

Step 4. The process ut F (Xt ) belongs to L1,2 because u ∈ L2,2 , v ∈ L1,2 , 1 4 E(DuL2 ([0,1]2 ) ) < ∞, and the processes F (Xt ), F (Xt ) and 0 u2s ds are uniformly bounded. In fact, we have Ds [ut F (Xt )] = ut Ft (Xt ) us 1{s≤t} + Ds X0 t t + Ds ur dW Wr + Ds vr dr + F (Xt )Ds ut , 0

0

and all terms in the right-hand side of the above expression are square integrable. For the third term we use the duality relationship of the Skorohod integral: 2 1 1 t E ut Ds ur dW Wr dsdt 0

0

0

& t = E Ds ur 2ut Dr ut Ds ur dW Wr + u2t Ds ur 0 0 0 0 ' t 2 +ut Dr Ds uθ dW Wθ drdsdt 0

4 ≤ cE DuL2 ([0,1]2 ) + D2 u L2 ([0,1]3 ) . 1

1

1

Step 5. Using the duality relationship it is clear that for any smooth random variable G ∈ S we have 2n −1 n t ti+1 n lim E G F (X(ti ))us dW Ws = E G F (Xs )us dW Ws . n→∞

i=0

tn i

0

On the other hand, we have seen that that An converges in L1 (Ω) to (3.33). Hence, (3.26) holds. Remarks: 1. If the process X is adapted then D− X = 0, and we obtain the classical Itˆˆo’s formula. 2. Also using the operator ∇ introduced in (3.8) we can write t 1 t F (Xt ) = F (X0 ) + F (Xs )dXs + F (Xs )(∇X)s us ds. 2 0 0

3.2 Stochastic calculus for anticipating integrals

189

3. A suﬃcient condition for X to have continuous paths is 1 p E Dut H dt < ∞ 0

for some p > 2 (see Proposition 3.2.2). 4. One can show Theorem 3.2.2 under diﬀerent types of hypotheses. More precisely, one can impose some conditions on X and modify the assumptions on X0 , u and v. For instance, one can assume either 1,2 (a) u ∈ L1,2 ∩ L∞ ([0, 1] × Ω), v ∈ L2 ([0, 1] × Ω), X ∈ L2− , and X has a version which is continuous and Xt ∈ D1,2 for all t, or 1,2 (b) u ∈ L1,2 ∩ L4 (Ω; L2 ([0, 1])), v ∈ L2 ([0, 1] × Ω), X ∈ L2− , and 1,2 ∈ D for all t, and X has a version which is continuous and X t 1 − 2 11 2 4 (D X ) dsdt + (D X) dt ∈ L (Ω). s t t 0 0 0

In fact, if X has continuous paths, by means of a localization argument it suﬃces to show the result for each function Fk , which has compact support. 1,2 On the other hand, the properties u ∈ L1,2 , v ∈ L2 ([0, 1]×Ω) and X ∈ L2− imply the convergences (3.28), (3.30) and (3.32). The boundedness or integrability assumptions on u, DX and D− X are used in order to ensure that E(Φ2t ) < ∞, and ut F (Xt ) ∈ L1,2 . 1,2 2,2 1,2 5. If we assume the conditions X0 ∈ Dloc , u ∈ Lloc and v ∈ Lloc , and X has continuous paths, then we can conclude that us F (Xs )1[0,t] (s) ∈ (Domδ)loc and Ito’s ˆ formula (3.26) holds. In fact, steps 1, 2 and 3 of the proof are still valid. Finally, the sequence of processes

n

v :=

n 2 −1

F (X(tni ))us 1(tni ,tni+1 ] (s)

i=0

converges in L ([0, 1] × Ω) to F (Xs )us 1[0,t] (s) as n tends to inﬁnity, and by Step 3, δ(v n ) converges in L1 (Ω) to Φt . Then, by Proposition 1.3.6, us F (Xs )1[0,t] (s) belongs to the domain of the divergence and (3.26) holds. 2

In [10] the following version of Itˆ oˆ’s formula is proved: t t Theorem 3.2.3 Suppose that Xt = X0 + 0 us dW Ws + 0 vs ds, where X0 ∈ 1,2 1,2,f Dloc , u ∈ LF ∩ L∞ (Ω; L2 ([0, 1])) loc , v ∈ Lloc and the process X has continuous paths. Let F : R → R be a twice continuously diﬀerentiable function. Then F (Xs )us 1[0,t] (s) belongs to (Domδ)loc and we have t 1 t F (Xt ) = F (X0 ) + F (Xs )dXs + F (Xs )u2s 2 0 0 t + F (Xs )(D− X)s us ds. (3.34) 0

190

3. Anticipating stochastic calculus

Proof: By a localization argument we can assume 1 that the processes F (Xt ), F (Xt ), F (Xt ) are uniformly bounded and 0 u2s ds ≤ k. Then we proceed as in the steps 1, 2 and 3 of the proof of Theorem 3.2.2, and we conclude using Proposition 1.3.6. Notice that for the proof of the convergence (3.28) we have to apply Theorem 3.2.1 for u ∈ LF . Also, (3.31) holds thanks to Proposition 1.3.5 applied to the set A = [tni , tni+1 ]. Remarks: t 1. A suﬃcient condition for the indeﬁnite Skorohodintegral 0us dW Ws of 1 F p a process u ∈ L to have a continuous version is E 0 |ut | dt < ∞ for some p > 2 (see Exercise 3.2.12). 1 2. The fact that 0 u2t dt is bounded is only used to insure that the righthand side of (3.26) is square integrable, and it can be replaced by the 1 1 2 conditions 0 u2t dt, 0 (D− X)t dt ∈ L2 (Ω). 3. 1

If X0 is a constant 1 2 and u and v are adapted processes such that < ∞ and 0 vt dt < ∞ a.s., then the these processes satisfy hypotheses of Theorem 3.2.3 because u ∈ LF ∩ L∞ (Ω; L2 ([0, 1])) loc by 1,2,f (this property can be proved by a localizaProposition 1.3.18, v ∈ Lloc tion procedure similat to that used in the proof of Proposition 1.3.18), and t t Ws + 0 vs ds has continuous paths because the process Xt = X0 + 0 us dW it is a continuous semimartingale. Furthermore, in this case D− X vanishes and we obtain the classical Itˆ o’s formula. u2 dt 0 t

1,2,f 4. In Theorem 3.2.3 we can replace the condition v ∈ Lloc by the fact 1 n ∞ |v | dt ∈ L (Ω), and that we can localize v by processes such that t 0 4 5 t n 1,2,f n Vt = 0 vs ds, t ∈ [0, 1] ∈ L2− . In this way the change-of-variable formula established in Theorem 3.2.3 is a generalization of Itˆ o’s formula.

The following result is a multidimensional and local version of the changeof-variables formula for the Skorohod integral. Theorem 3.2.4 Let W = {W Wt , t ∈ [0, 1]} be an N -dimensional Brownian 2,2 , and v i ∈ L1,2 , 1 ≤ i ≤ M , motion. Suppose that X0i ∈ D1,2 , uij ∈ L(4) 1 ≤ j ≤ N are processes such that Xti

=

X0i

+

N j=1

0

t

uij Wsj s dW

t

vsi ds, 0 ≤ t ≤ 1.

+ 0

3.2 Stochastic calculus for anticipating integrals

191

Assume that Xti has continuous paths. Let F : RM → R be a twice continuously diﬀerentiable function. Then we have F (Xt ) = F (X0 ) +

M

t

(∂ ∂i F )(Xs )dXsi

0

i=1

M N 1 t jk + (∂ ∂i ∂j F )(Xs )uik s us ds 2 i,j=1 0 k=1

M

+

N t

i,j=1 k=1

(∂ ∂i ∂j F )(Dk,− X j )s uik s ds,

0

where Dk denotes the derivative with respect to the kth compoment of the Wiener process. Ito’s ˆ formula for the Skorohod integral allows us to deduce a change-ofvariables formula for the Stratonovich integral. Let us ﬁrst introduce the following classes of processes: 2,2 continuous in L2 (Ω) The set LS2,4 is the class of processes u ∈ L11,2 ∩ L(4) and such that ∇u ∈ L1,2 . Theorem 3.2.5 Let F be a real-valued, twice continuously diﬀerentiable t t function. Consider a process of the form Xt = X0 + 0 us ◦ dW Ws + 0 vs ds, 1,2 2,4 1,2 , u ∈ LS,loc and v ∈ Lloc . Suppose that X has continuous where X0 ∈ Dloc paths. Then we have F (Xt ) = F (X0 ) +

t

F (Xs )vs ds +

0

t

[F (Xs )us ] ◦ dW Ws .

0

Proof: As in the proof of the change-of-variable formula for the Skorohod integral we can assume that the processes F (Xt ), F (Xt ), F (Xt ) and 1 2 u ds are uniformly bounded, X0 ∈ D1,2 , u ∈ LS2,4 and v ∈ L1,2 . We know, 0 s by Theorem 3.1.1 that the process Xt has the following decomposition: Xt = X0 +

t

us dW Ws + 0

t

vs ds + 0

0

t

1 (∇u)s ds. 2

This process veriﬁes the assumptions of Theorem 3.2.2. Consequently, we can apply Itˆ o’s formula to X and obtain F (Xt )

t 1 t = F (X0 ) + F (Xs )vs ds + F (Xs )(∇u)s ds 2 0 0 t 1 t F (Xs )us dW Ws + F (Xs )(∇X)s us ds. + 2 0 0

192

3. Anticipating stochastic calculus

The process F (Xt )ut belongs to L11,2 . In fact, notice ﬁrst that as in the 1 proof of Theorem 3.2.2 the boundedness of F (Xt ), F (Xt ) and 0 u2s ds 2,2 , v, ∇u ∈ L1,2 and X0 ∈ D1,2 imply that this and the fact that u ∈ L(4) 1,2 process belongs to L and Ds [F (Xt )ut ] = F (Xt )Ds ut + F (Xt )Ds Xt ut . On the other hand, using that u ∈ L11,2 , u is continuous in L2 (Ω), and X ∈ L21,2 we deduce that F (Xt )ut belongs to L11,2 and that (∇ (F (X)u))t = F (Xt )(∇u)t + F (Xt )ut (∇X)t . Hence, applying Theorem 3.1.1, we can write t t 1 t [F (Xs )us ] ◦ dW Ws = F (Xs )us dW Ws + (∇ (F (X)u))s ds 2 0 0 0 t 1 t F (Xs )us dW Ws + F (Xs )(∇u)s ds = 2 0 0 t 1 F (Xs )us (∇X)s ds. + 2 0 Finally, notice that

(∇X)t = 2(D− X)t + ut .

This completes the proof of the theorem. In the next theorem we state a multidimensional version of the changeof-variable formula for the Stratonovich integral. Theorem 3.2.6 Let W = {W Wt , t ∈ [0, 1]} be an N -dimensional Brownian motion. Suppose that X0i ∈ D1,2 , uij ∈ LS2,4 , and v i ∈ L1,2 , 1 ≤ i ≤ M , 1 ≤ j ≤ N are processes such that Xti = X0i +

N

uij Wsj + s ◦ dW

0

j=1

t

t

vsi ds, 0 ≤ t ≤ 1. 0

Assume that Xti has continuous paths. Let F : RM → R be a twice continuously diﬀerentiable function. Then we have F (Xt ) = F (X0 ) +

M N i=1 j=1

0

t

(∂ ∂i F )(Xs )uij Wsj + s ◦ dW

M i=1

t

(∂ ∂i F )(Xs )vsi ds.

0

Similar results can be obtained for the forward and backward stochastic integrals. In these cases we require some addtional continuity conditions. Let us consider the case of the forward integral. This integral is deﬁned as the limit in probability of the forward Riemann suns:

3.2 Stochastic calculus for anticipating integrals

193

Deﬁnition 3.2.1 Let u = {ut , t ∈ [0, 1]} be a stochastic process. The for1 ward stochastic integral 0 ut d− Wt is deﬁned as the limit in probability as |π| → 0 of the Riemann sums n−1

uti (W Wti+1 − Wti ).

i=1

The following proposition provides some suﬃcient conditions for the existence of this limit. Proposition 3.2.3 Let u = {ut , t ∈ [0, 1]} be a stochastic process which is 1,2 . Then u continuous in the norm of the space D1,2 . Suppose that u ∈ L1− is forward integrable and 1 1 − − δ(u) = D u s ds. u t d Wt + 0

0

Proof: We can write, using Eq. (3.4) n−1 n−1 n−1 δ uti 1(ti ,ti+1 ] = uti (W Wti+1 − Wti ) − i=1

i=1

i=1

ti+1

ti

Ds uti ds. (3.35)

The processes uπ− = t

n−1

uti 1(ti ,ti+1 ] (t)

i=1

converge in the norm of the space L1,2 to the process u, due to the continuity of t → ut in D1,2 . Hence, left-hand side of Equation (3.35), which equals to δ(uπ− ), converges in L2 (Ω) to δ(u). On the other hand, n−1 1 ti+1 − E D u s ds Ds uti ds − 0 i=1 ti n−1 ti+1 ≤ E Ds uti − D− u s ds i=1 1

≤

ti

sup

E(|Ds Xt − (D− X)s |)ds,

0 (s−|π|)∨0≤t<s

which tends to zero as |π tends to zero, by the deﬁnition of the space 1,2 L1− . We can establish an Itˆ o’s formula for the forward integral as in the case of the Stratonovich integral. Notice that the forward integral follows the rules 2,4 as the class of processes of the Itˆ ˆo stochastic calculus. Deﬁne the set L− 1,2 2,2 1,2 u ∈ L1− ∩ L(4) continuous in D and such that D− u ∈ L1,2 .

194

3. Anticipating stochastic calculus

Theorem 3.2.7 Let F be a real-valued, twice continuously diﬀerentiable t t function. Consider a process of the form Xt = X0 + 0 us d− Ws + 0 vs ds, 1,2 2,4 1,2 , u ∈ L−, where X0 ∈ Dloc loc and v ∈ Lloc . Suppose that X has continuous paths. Then we have t t 1 t − F (Xt ) = F (X0 )+ F (Xs )vs ds+ [F (Xs )us ]d Ws + F (Xs )u2s ds. 2 0 0 0 (3.36) Proof: As in the proof of the change-of-variable formula for the Skorohod integral we can assume that the processes F (Xt ), F (Xt ), F (Xt ) and 1 2 2,4 u ds are uniformly bounded, X0 ∈ D1,2 , u ∈ L− and v ∈ L1,2 . We know, 0 s by Proposition 3.2.3 that the process Xt has the following decomposition: t t t Xt = X0 + us dW Ws + vs ds + (D− u)s ds. 0

0

0

This process veriﬁes the assumptions of Theorem 3.2.2. Consequently, we can apply Itˆ o’s formula to X and obtain t t F (Xt ) = F (X0 ) + F (Xs )vs ds + F (Xs )(D− u)s ds 0 0 t 1 t F (Xs )us dW Ws + F (Xs )(∇X)s us ds. + 2 0 0 1,2 The process F (Xt )ut belongs to L1− . In fact, notice ﬁrst that as in the 1 proof of Theorem 3.2.2 the boundedness of F (Xt ), F (Xt ) and 0 u2s ds 2,2 and the fact that u ∈ L(4) , v, D− u ∈ L1,2 and X0 ∈ D1,2 imply that this 1,2 process belongs to L and

Ds [F (Xt )ut ] = F (Xt )Ds ut + F (Xt )Ds Xt ut . 1,2 On the other hand, using that u ∈ L1− , u is continuous in L2 (Ω), X has 1,2 1,2 continuous paths, and X ∈ L2 we deduce that F (Xt )ut belongs to L1− and that − D (F (X)u) t = F (Xt )(D− u)t + F (Xt )ut (D− X)t .

Hence, applying Proposition 3.2.3, we can write t t t − D (F (X)u) s ds [F (Xs )us ]d− Ws = F (Xs )us dW Ws + 0 0 0 t t F (Xs )us dW Ws + F (Xs )(D− u)s ds = 0 0 t F (Xs )us (D− X)s ds. + 0

3.2 Stochastic calculus for anticipating integrals

195

Finally, notice that (∇X)t = 2(D− X)t + ut .

This completes the proof of the theorem.

3.2.4

Substitution formulas

The aim of this section is to study the following problem. Suppose that u = {ut (x), 0 ≤ t ≤ 1} is a stochastic process parametrized by x ∈ Rm , which is square integrable and adapted for each x ∈ Rm . For each x we can deﬁne the Itˆo integral 1 ut (x)dW Wt . 0

Assume now that the resulting random ﬁeld is a.s. continuous in x, and let F be an m-dimensional random variable. Then we can evaluate the stochastic integral at x = F , that is, we can deﬁne the random variable

1

ut (x)dW Wt |x=F .

(3.37)

0

A natural question is under which conditions is the nonadapted process {ut (F ), 0 ≤ t ≤ 1} Skorohod integrable, and what is the relationship between the Skorohod integral of this process and the random variable deﬁned by (3.37). We will show that this problem can be handled if the random ﬁeld ut (x) is continuously diﬀerentiable in x and veriﬁes some integrability conditions, and, on the other hand, the random variable F belongs locally to D1,4 . Notice, however, that no kind of smoothness in the sense of the Malliavin calculus will be required on the process ut (x). A similar question can be asked for the Stratonovich integral. To handle these problems we will make use of the following technical result. Lemma 3.2.2 Suppose that {Y Yn (x), x ∈ Rm }, n ≥ 1 is a sequence of random ﬁelds such that Yn (x) converges in probability to Y (x) as n tends to inﬁnity, for each ﬁxed x ∈ Rm . Suppose that E(|Y Yn (x) − Yn (y)|p ) ≤ cK |x − y|α ,

(3.38)

for all |x|, |y| ≤ K, n ≥ 1, K > 0 and for some constants p > 0 and α > m. Then, for any m-dimensional random variable F one has lim Yn (F ) = Y (F )

n→∞

in probability. Moreover, the convergence is in Lp (Ω) if F is bounded.

196

3. Anticipating stochastic calculus

Proof: Fix K > 0. Replacing F by FK := F 1{|F |≤K} we can assume that F is bounded by K. Fix ε > 0 and consider a random variable Fε which takes ﬁnitely many values and such that |F Fε | ≤ K and F − Fε ∞ ≤ ε. We can write |Y Yn (F ) − Y (F )| ≤ |Y Yn (F ) − Yn (F Fε )| + |Y Yn (F Fε ) − Y (F Fε )| + |Y (F Fε ) − Y (F )|. Take 0 < m < α − m. By (3.38) and (A.10) there exist a constant C1 and random variables Γn such that |Y Yn (x) − Yn (y)|p E(Γn )

≤

|x − y|m Γn ,

≤ C1 .

Moreover, Eq. (3.38) is also satisﬁed by the random ﬁeld Y (x) and we can also ﬁnd a constant C2 and a random variable Γ such that |Y (x) − Y (y)|p E(Γ) Hence, E (|Y Yn (F ) − Y (F )|p ) ≤ cp

≤ |x − y|m Γ, ≤ C2 .

(C1 + C2 )εm + E (|Y Yn (F Fε ) − Y (F Fε )|p ) ,

and the desired convergence follows by taking ﬁrst the limit as n tends to inﬁnity and then the limit as ε tends to zero. Consider a random ﬁeld u = {ut (x), 0 ≤ t ≤ 1, x ∈ Rm } satisfying the following conditions: (h1) For each x ∈ Rm and t ∈ [0, 1], ut (x) is Ft -measurable (h2) There exist constants p ≥ 2 and α > m such that E(|ut (x) − ut (y)|p ) ≤ Ct,K |x − y|α , 1 for all |x|, |y| ≤ K, K > 0, where 0 Ct,K dt < ∞. Moreover 1 E(|ut (0)|2 )dt < ∞. 0 Notice that under the above conditions for each t ∈ [0, 1] the random ﬁeld {ut (x), x ∈ Rm } possesses a continuous version, and the Itˆo integral 1 u (x)dW Wt possesses a continuous version in (t, x). In fact, for all |x|, 0 t |y| ≤ K, K > 0, we have t p (us (x) − us (y)) dW Ws E sup t∈[0,1]

0

p2 t 2 ≤ cp E |us (x) − us (y)| ds 0 1 ≤ cp Ct,K dt |x − y|α . 0

3.2 Stochastic calculus for anticipating integrals

197

The following theorem provides the relationship between the evaluated 1 integral 0 ut (x)dW Wt and the Skorohod integral δ(u(F )). We need the x=F

following hypothesis which is stronger than (h2): (h3) For each (t, ω) the mapping x → ut (x) is continuously diﬀerentiable, and for each K > 0 1

sup |∇ut (x)|q

E

dt < ∞,

|x|≤K

0

1

where q ≥ 4 and q > m. Moreover

E(|ut (0)|2 )dt < ∞.

0

Theorem 3.2.8 Suppose that u = {ut (x), 0 ≤ t ≤ 1, x ∈ Rm } is a random ﬁeld satisfying conditions (h1) and (h3). Let F : Ω → Rm be a bounded random variable such that F i ∈ D1,4 for 1 ≤ i ≤ m. Then the composition u(F ) = {ut (F ), 0 ≤ t ≤ 1} belongs to the domain of δ and δ(u(F )) = 0

Proof:

1

ut (x)dW Wt

−

m

x=F

j=1

1

∂j ut (F )Dt F j dt.

(3.39)

0

Consider the approximation of the process u given by unt (x)

=

n−1

i n

n

−1 n

i=1

us (x)ds 1( i , i+1 ] (t). n

(3.40)

n

Notice that u(F ) belongs to L2 ([0, 1] × Ω) and the sequence of processes un (F ) converges to u(F ) in L2 ([0, 1] × Ω) as n tends to inﬁnity. Indeed, if F is bounded by K we have E

1

2

ut (F ) dt

1

≤

E

0

2

sup |ut (x)| dt

|x|≤K

0

1

≤ 2 0

+2K

E(|ut (0)|2 )dt 1

2

E 0

2

sup |∇ut (x)| dt

|x|≤K

< ∞. The convergence of un (F ) to u(F ) in L2 ([0, 1] × Ω) is obtained by ﬁrst approximating u(F ) by an element of C([0, 1]; L2 (Ω)).

198

3. Anticipating stochastic calculus

From Proposition 1.3.4 and Exercise 3.2.10 we deduce that un (F ) belongs to Domδ and i m n−1 n−1 n n W i+1 − W ni − δ(u (F )) = n us (F )ds n −1 n

i=1

×

i n −1 n

The Itˆo stochastic integrals E

1

n

i+1 n

∂j us (F )ds

i=1 j=1

Dr F j dr .

(3.41)

i n

1

unt (x)dW Wt satisfy

0

unt (x)dW Wt

−

0

1

0

q

unt (y)dW Wt

≤ cq |x − y|q .

1 Hence, by Lemma 3.2.2 the sequence 0 unt (x)dW Wt converges in Lq (Ω) x=F 1 to the random variable 0 ut (x)dW Wt . On the other hand, the second x=F summand in the right-hand side of (3.41) converges to m

1

∂j ut (F )Dt F j dt

0

j=1

in L2 (Ω), as it follows from the estimate

1

1

∂j ut (F )Dt F j dt −

0

≤ DF j H

0 1

∂j unt (F )Dt F j dt 2

|∂ ∂j ut (F ) − ∂j unt (F )| dt

12 .

0

The operator δ being closed the result follows.

The preceding theorem can be localized as follows: Theorem 3.2.9 Suppose that u = {ut (x), 0 ≤ t ≤ 1, x ∈ Rm } is a random ﬁeld satisfying conditions ((h1) and (h3). Let F : Ω → Rm be a random 1,4 for 1 ≤ i ≤ m. Then the composition u(F ) = variable such that F i ∈ Dloc {ut (F ), 0 ≤ t ≤ 1} belongs to (Domδ)loc and δ(u(F )) = 0

1

ut (x)dW Wt

− x=F

m j=1

1

∂j ut (F )Dt F j dt.

(3.42)

0

We recall that the operator δ is not known to be local in Dom δ, and for this reason the value of δ(u(F )) in the Theorem 3.2.9 might depend on the particular localizing sequence.

3.2 Stochastic calculus for anticipating integrals

199

The Stratonovich integral also satisﬁes a commutativity relationship but in this case we do not need a complementary term. This fact is coherent with the the general behavior of the Stratonovich integral as an ordinary integral. Let us introduce the following condition which is stronger than (h2): (h4) There exists constants p ≥ 2 and α > m such that E(|ut (x) − ut (y)|p ) E(|ut (x) − ut (y) − us (x) + us (y)|p )

≤ cK |x − y|α , p

≤ cK |x − y|α |t − s| 2 ,

1 for all |x|, |y| ≤ K, K > 0, and s, t ∈ [0, 1]. Moreover 0 E(|ut (0)|2 ) dt < ∞, and t → ut (x) is continuous in L2 (Ω) for each x. Theorem 3.2.10 Let u = {ut (x), 0 ≤ t ≤ 1, x ∈ Rm } be a random ﬁeld satisfying hypothesis (h1) and (h4). Suppose that for each x ∈ Rm the 1 Wt exists. Consider an arbitrary random Stratonovich integral 0 ut (x) ◦ dW variable F . Then u(F ) is Stratonovich integrabl, and we have

1

ut (F ) ◦ dW Wt =

0

S

0

ut (x) ◦ dW Wt

.

(3.43)

x=F

Fix a partition π = {0 = t0 < t1 < · · · < tn = 1}. We can write

Proof: π

1

=

n−1 i=0

=

n−1

1 ti+1 − ti

us (F )ds (W (ti+1 ) − W (ti ))

ti

uti (F )(W (ti+1 ) − W (ti ))

i=0

+

ti+1

n−1 i=0

1 ti+1 − ti

ti+1

ti

[us (F ) − uti (F )]ds (W (ti+1 ) − W (ti ))

= an + bn . The continuity of t → ut (x) in L2 (Ω) implies that for each x ∈ Rm the Riemann sums n−1 uti (x)(W (ti+1 ) − W (ti )) i=0

converge in L2 (Ω), as |π| tends to zero, to the Ito ˆ integral Moreover, we have for |x|, |y| ≤ K,

1 0

ut (x)dW Wt .

p n−1 [uti (x) − uti (y)] (W (ti+1 ) − W (ti )) ≤ cp cK |x − y|α . E i=0

200

3. Anticipating stochastic calculus

Hence, by Lemma 3.2.2 we deduce that an converges in probability to 1 u (x) ◦ dW W . Now we study the convergence of bn . For each ﬁxed t 0 t x=F m x ∈ R , the sums Rπ (x) :=

n−1 i=0

1 ti+1 − ti

ti+1

ti

[us (x) − uti (x)]ds (W (ti+1 ) − W (ti ))

converge in probability, as |π| tends to zero, to the diﬀerence 1 1 V (x) := ut (x) ◦ dW Wt − ut (x)dW Wt . 0

0

So, it suﬃces to show that R (F ) converges in probability, as |π| tends to zero, to V (F ). This convergence follows from Lemma 3.2.2 and the following estimates, where 0 < ε < 1 veriﬁes εα > m, and |x|, |y| ≤ K π

Rπ (x) − Rπ (x)pε ≤

n−1 i=0

×

1 ti+1 − ti ti+1

ti

≤ cp,ε

n−1

[us (x) − uti (x) − us (y) + uti (y)](W (ti+1 ) − W (ti ))pε ds 1

|ti+1 − ti | 2

sup s∈[ti ,ti+1 ]

i=0

p

1

[E (|us (x) − uti (x) − us (y) + uti (y)| )] p

α p

≤ cp,ε cK |x − y| . It is also possible to establish substitution formulas similar to those appearing in Theorems 3.2.9 and 3.2.10 for random ﬁelds ut (x) which are not adapted. In this case additional regularity assumptions are required. More precisely we need regularity in x of Ds ut (x) for Theorem 3.2.9 and of (∇u(x))t for Theorem 3.2.10. One can also compute the diﬀerentials of processes of the form Ft (Xt ) Ft (x), t ∈ [0, 1], x ∈ Rm } are generalized continwhere {Xt , t ∈ [0, 1]} and {F uous semimartingales, i.e., they have a bounded variation component and an indefnite Skorohod (or Stratonovich) integral component. This type of change-of-variables formula are known as Itˆˆo-Ventzell formulas. We show below a formula of this type that will be used to solve anticipating stochastic diﬀerential equations. We ﬁrst state the following diﬀerentiation rule (see also Exercise 1.3.6). Lemma 3.2.3 Let F = (F 1 , . . . , F m ) be a random vector whose components belong to D1,p , p > 1 and suppose that |F | ≤ K. Consider a measurable random ﬁeld u = {u(x), x ∈ Rm } with continuously diﬀerentiable

3.2 Stochastic calculus for anticipating integrals

201

paths, such that for any x ∈ Rm , u(x) ∈ D1,r , r > 1, and the derivative Du(x) has a continuous version as an H-valued process. Suppose we have E

r

sup [|u(x)|r + Du(x)H ]

|x|≤K

sup |∇u(x)|q

|x|≤K 1 p

+

1 q

(3.44)

< ∞,

(3.45)

E where

< ∞,

= 1r . Then the composition u(F ) belongs to D1,r , and we have D (u(F )) =

m

∂i u(F )DF i + (Du)(F ).

(3.46)

i=1

Proof:

Consider an approximation of the identity ψ n , and deﬁne un (x) = u(y)ψ n (x − y)dy. Rm

The sequence of random variables un (F ) converges almost surely and in in Lr (Ω) to u(F ) because of Condition (3.44). On the other hand, un (F ) ∈ D1,r and D (un (F )) = D u(y)ψ n (F − y)dy = Rm

Rm

Du(y)ψ n (F − y)dy +

Rm

Rm

i=1

=

m

Du(y)ψ n (F − y)dy +

m

u(y)∂ ∂i ψ n (F − y)DF i dy

DF i Rm

i=1

∂i u(y)ψ n (F − y)dy.

Again by conditions (3.44) and (3.45), this expression converges in Lr (Ω; H) to the right-hand side of (3.46). Let W be an N -dimensional Wiener process. Consider two stochastic processes of the form: Xti = X0i +

N j=1

t

t

uij Wsj + s ◦ dW

0

vsi ds, 1 ≤ i ≤ m 0

and Ft (x) = F0 (x) +

N j=1

t

Hsj (x) ◦ dW Wsj +

0

We introduce the following set of hypotheses:

t

Gs (x)ds, x ∈ Rm . 0

202

3. Anticipating stochastic calculus

(A1) For all 1 ≤ j ≤ N , 1 ≤ i ≤ m, X0i ∈ D1,2 , uij ∈ LS2,4 , v i ∈ L1,2 , and the process X is continuous and bounded by a constant K. We assume also that u is bounded. Ft (x) (A2) x → Ft (x) is twice diﬀerentiable for all t; the processes Ft (x), ∇F and ∇2 Ft (x) are continuous in (t, x); F (x) ∈ L1,2 , and there exist two versions of Ds Ft (x) which are diﬀerentiable in x and such that the functions Ds Ft (x) and ∇(Ds Ft )(x) are continuous in the regions {s ≤ t ≤ 1, x ∈ Rm } and {t ≤ s ≤ 1, x ∈ Rm }, respectively. (A3) x → Htj (x) is diﬀerentiable for all t; the processes Ht (x) and ∇H Ht (x) are continuous in (t, x); H j (x) ∈ LS2,4 , and there exist two versions of Ds Ht (x) which are diﬀerentiable in x and such that the functions Ds Ht (x) and ∇(Ds Ht )(x) are continuous in the regions {s ≤ t ≤ 1, x ∈ Rm } and {t ≤ s ≤ 1, x ∈ Rm }, respectively. (A4) The function x → Gt (x) is continuous for all t, and G(x) ∈ L1,2 . (A5) The following estimates hold: 2 4 4 4 E sup |F Ft (x)| + |∇F < ∞, Ft (x)| + |∇ Ft (x)| |x|≤K,t∈[0,1]

E

sup |x|≤K,t∈[0,1]

|H Ht (x)| + |∇H Ht (x)| + |Gt (x)|

1

E

0

sup |x|≤K,t∈[0,1]

4

4

|Ds Ft (x)| + |∇(Ds Ft )(x)|

< ∞,

ds <

∞,

4 4 |Ds Ht (x)| + |∇(Ds Ht )(x)| ds <

∞.

|x|≤K,t∈[0,1]

E 0

sup

1

4

4

4

Theorem 3.2.11 @Assume that Athe processes Xt and Ft (x) satisfy the above hypotheses. Then ∇F Ft (Xt ), ujt and Htj (Xt ) are elements of L11,2 for all j = 1, . . . , N , and Ft (Xt ) = F0 (X0 ) +

N t # j=1

+

N j=1

0

$ ∇F Fs (Xs ), ujs ◦ dW Wsj +

0

t

∇F Fs (Xs ), vs ds 0

t

Hsj (Xs ) ◦ dW Wsj +

t

Gs (Xs )ds. 0

Proof: To simplify then notation we will assume N = m = 1. The proof will be done in several steps. Step 1: Consider an approximation of the identity ψ n on R. For each x ∈ R we can apply the multidimensional change-of-variables formula for

3.2 Stochastic calculus for anticipating integrals

203

the Stratonovich integral (Theorem 3.2.6) to the process Ft (x)ψ n (Xt − x) and we obtain t Hs (x)ψ n (Xs − x) ◦ dW Ws Ft (x)ψ n (Xt − x) = F0 (x)ψ n (X0 − x) + 0 t + Gs (x)ψ n (Xs − x)ds 0 t Fs (x)ψ n (Xs − x)us ◦ dW Ws + 0 t + Fs (x)ψ n (Xs − x)vs ds. 0

If we express the above Stratonovich integrals in terms of the corresponding Skorohod integrals we get t Hs (x)ψ n (Xs − x)dW Ws Ft (x)ψ n (Xt − x) = F0 (x)ψ n (X0 − x) + 0 ' t& 1 + Gs (x) + (∇H)s (x) ψ n (Xs − x)ds 2 0 t 1 + Hs (x)ψ n (Xs − x) (∇X)s ds 2 0 t Fs (x)ψ n (Xs − x)us dW Ws + 0 t 1 + (∇F )s (x)ψ n (Xs − x)us ds 2 0 t 1 (∇u)s + vs ds Fs (x)ψ n (Xs − x) + 2 0 1 t + Fs (x)ψ n (Xs − x)(∇X)s us ds. 2 0 Grouping the Skorohod integral terms and the Lebesgue integral terms together we can write Ft (x)ψ n (Xt − x) = F0 (x)ψ n (X0 − x) +

t

t

αn (s, x)dW Ws + 0

β n (s, x)ds, 0

where αn (t, x) = ψ n (Xt − x)H Ht (x) + ψ n (Xt − x)ut Ft (x)

204

3. Anticipating stochastic calculus

and 1 = ψ n (Xt − x) Gt (x) + (∇H)t (x) 2 1 Ht (x) (∇X)t +ψ n (Xt − x) 2 1 1 (∇u)t + vt + ut (∇F )t (x) +F Ft (x) 2 2

β n (t, x)

1 + ψ n (Xt − x)F Ft (x) (∇X)t ut . 2 Step 2: We claim that the processes Ht (Xt ) and Ft (Xt )ut belong to L11,2 . In fact, ﬁrst notice that Hypotheses (A2) and (A3) imply that for each x, Ft (x)) = (Ds Ft ) (x). Then we Ht (x) and Ft (x)ut belong to L1,2 , and Ds (F can apply Lemma 3.2.3 (suitably extended to stochastic processes) to Ht (x) and Xt and also to Ft (x)ut and Xt to deduce that Ht (Xt ) and Ft (Xt )ut belong to L1,2 , taking into account the following estimates 1 1 2 2 E(H (X ))ds ≤ E sup H (x) ds < ∞, s s 0 0 |x|≤K s 1 2 E sup|x|≤K DHs (x)H ds < ∞, 0 1 4 ds < ∞, E sup |H (x)| |x|≤K s 0 and 1 01 0 1

E((F Fs (Xs )us ) )ds ≤ u∞ 2

2

1 0

2 E sup|x|≤K |F Fs (x)| ds < ∞,

E(|F Fs (Xs )| Dus H )ds 12 1 1 4 4 ≤ 0 E sup|x|≤K |F Fs (x)| ds 0 E Dus H ds < ∞, 2

2

E(|F Fs (Xs )| Dus H u2s )ds ≤ u∞ 12 1 1 4 4 × 0 E sup|x|≤K |F Fs (x)| ds 0 E Dus H ds < ∞,

2 2 1 Fs ) (Xs ) H us )ds E( (DF 0

2

2 1 Fs ) (x) H ds < ∞. ≤ u∞ 0 E sup|x|≤K (DF 2

2

2

0

The derivatives of these processes are given by Ds [H Ht (Xt )] = Ht (Xt )Ds Xt + (Ds Ht )(Xt ) and Ds [F Ft (Xt )ut ] = Ft (Xt )Ds Xt ut + (Ds Ft ) (Xt )ut + Ft (Xt )Ds ut .

3.2 Stochastic calculus for anticipating integrals

205

Then, from our hypotheses it follows that the processes Ht (Xt ) and Ft (Xt )ut belong to L11,2 , and (∇ (H· (X· )))t = Ht (Xt ) (∇X)t + (∇H)t (Xt ) and (∇(F· (X· )u· ))t = Ft (Xt ) (∇X)t ut + (∇F ) (Xt )ut + Ft (Xt ) (∇u)t . Step 3: R

We claim that Ft (x)ψ n (Xt − x) − F0 (x)ψ n (X0 − x) −

t

β n (s, x)ds dx

(3.47)

0

converges in L1 (Ω) to the process Φt

t& 1 = Ft (Xt ) − F0 (X0 ) − Gs (Xs ) + (∇H)s (Xs ) 2 0 1 1 + Hs (Xs ) (∇X)s + (∇F )s (Xs )us + Fs (Xs ) (∇u)s 2 2 +F Fs (Xs ) (∇X)s us + Fs (Xs )vs ] ds.

In fact, the convergence for each (s, ω) follows from the continuity assumption on x of the processes Gs (x), (∇H)s (x), (∇F )s (x), and the fact that Hs (x) is of class C 1 , and Fs (x) is of class C 2 . The L1 (Ω) convergence follows by the dominated convergence theorem because we have 1 sup |β n (s, x)| dxds < ∞. E n

0

R

From Step 2 we deduce

Φt

t

= Ft (Xt ) − F0 (X0 ) − [Gs (Xs ) + Fs (Xs )vs ] ds 0 t [Hs (Xs ) + Fs (Xs )us ] ◦ dW Ws − 0 t + [Hs (Xs ) + Fs (Xs )us ] dW Ws . 0

Step 4: Finally, we have αn (s, x)dx = Hs (x)ψ n (Xs − x)dx + Fs (x)ψ n (Xs − x)us dx. R

R

R

The integrals R αn (s, x)dx converge as n tends to inﬁnity in the norm of L1,2 to Hs (Xs ) + Fs (Xs )us . Hence, the Skorohod integral t αn (s, x)dx dW Ws (3.48) 0

R

206

3. Anticipating stochastic calculus

converges in L2 (Ω) to

t

[Hs (Xs ) + Fs (Xs )us ] dW Ws .

0

By Fubini’s theorem, (3.48) coincides with (3.47). Consequently, we get t Φt = [Hs (Xs ) + Fs (Xs )us ] dW Ws , 0

which completes the proof.

Exercises 3.2.1 Consider the process u deﬁned by u(t) = 1{W (1)>0} (1[0, 12 ] (t) − 1( 12 ,1] (t)). Show that this process is Skorohod integrable but the process u1[0, 12 ] is not. 3.2.2 Let u ∈ L1,2 , and ﬁx 0 ≤ s < t ≤ 1. Show that 2 t E ur dW Wr |F F[s,t]c s

t

t

u2r dr

=E s

+ s

t

s

Dv ur Dr uv drdv|F F[s,t]c

.

3.2.3 Consider the process u(t) = sign(W W1 − t) exp(W Wt − 2t ), 0 ≤ t ≤ 1. Show that this process is Skorohod integrable on any interval, and t t W1 . us dW Ws = sign(W W1 − t) exp(W Wt − ) − signW 2 0 1 Hint: Fix a smooth random variable G and compute E( 0 Dt Gut dt) using Girsanov’s theorem. 3.2.4 Let u ∈ L2,2 be a process satisfying ' & 1 sup |E(Ds ut )| + E |Ds Dr ut |2 dr < ∞. s,t∈[0,1]

0

1 Assume in addition that E 0 |ut |p dt < ∞ for some p > 4. Show that the t Skorohod integral { 0 us dW Ws , t ∈ [0, 1]} possesses a continuous version. 3.2.5 Let u ∈ L2 ([0, 1] × Ω) be a stochastic process such that for any t ∈ [0, 1] the random variable ut belongs to the ﬁnite sum of chaos ⊕N n=0 Hn .

3.2 Stochastic calculus for anticipating integrals

207

t Show that the Skorohod integral { 0 us dW Ws , t ∈ [0, 1]} possesses a continuous version. Hint: Use the fact that all the p norms are equivalent on a ﬁnite chaos and apply a deterministic change of time (see Imkeller [148]). 3.2.6 Let u be a process in the space L2,4 = L4 ([0, 1]; D2,4 ). Using the Gaussian formula (A.1) and Proposition 3.2.1, prove the following inequality: ( 2 2 2 1 1 1 1 2 Dr uθ dW Wθ dr ≤ C E (Dr uθ ) drdθ E 0

0

1

1

+E 0

0

0

1

2 ) 2 (Dα Dr uθ ) dαdr dθ .

0

0

3.2.7 Consider a random ﬁeld {ut (x), 0 ≤ t ≤ 1, x ∈ G}, where G is a bounded open subset of Rm , such that u ∈ L2 (G × [0, 1] × Ω). Suppose that for each x ∈ Rm , u(x) ∈ Dom δ and E( G |δ(u(x))|2 dx) < ∞. Show that the process { G ut (x)dx, t ∈ [0, 1]} is Skorohod integrable and δ ut (x)dx = δ(u· (x))dx. G

G

Yt , t ∈ [0, 1]} be continuous 3.2.8 Let X = {Xt , t ∈ [0, 1]} and Y = {Y 1,2 such that X is adapted and Y is backward processes in the space Lloc adapted (that is, Yt is F[t,1] -measurable for any t). Consider a C 1 function 1,2 Φ : R2 → R. Show that the process {Φ(Xt , Yt ), t ∈ [0, 1]} belongs to Lloc and that the sums n−1

Φ(X(ti ), Y (ti+1 ))(W (ti+1 ) − W (ti ))

i=0

converge in probability to the Skorohod integral doux and Protter [281]).

1 0

Φ(Xt , Yt )dW Wt (see Par-

3.2.9 Let f, g : R → R be bounded functions of bounded variation. Consider the stochastic process X(t) = f (W (t))g(W (1) − W (t)). Show that X is Skorohod integrable, and compute E(δ(X)2 ). 3.2.10 Suppose that W = {W (h), h ∈ H = L2 (T, B, µ)} is a centered isonormal Gaussian process on the complete probability space (Ω, F, P ). Assume F = σ(W ). Let G be a bounded domain of Rm with Lipschitz boundary, and let A ∈ B, µ(A) < ∞. Let {u(x), x ∈ G} be an FAc measurable random ﬁeld with continuously diﬀerentiable paths such that 4 2 E sup |∇u(x)| < ∞ and E sup |u(x)| < ∞. x∈G

x∈G

208

3. Anticipating stochastic calculus

Set h = 1A and let F ∈ D1,4 . Show that

m (a) u(F ) ∈ D2,h and Dh (u(F )) = i=1 ∂i uDh F i . (b) u(F )1A belongs to the domain of δ and δ(u(F )1A ) = u(F )W (A) −

m

∂i u(F )Dh F i .

i=1

Hint: Approximate u(F ) by convolution with an approximation of the identity. 2p 3.2.11 Let p ∈ (2, 4), and α = 4−p . Consider a process u = {ut , t ∈ [0, 1]} F α in L ∩ L ([0, 1] × Ω).Using the Itˆ oˆ’s formula for the Skorohod integral show the following estimate p t t p −1 2 ur dW Wr ≤ Cp (t − s) |ur |α dr+ E E s

s

t

t

2

+E

(Dr uθ ) drdθ t t t 2 (Dσ Dr uθ ) drdσdθ . +E s

θ

s

θ

θ

1 3.2.12 Let u = {ut , t ∈ [0, 1]} be a process in LF such that E 0 |ut |p dt < t ∞ for some p > 2. Show that the process Xt = 0 us dW Ws has a continuous version whose trajectories are H¨ o¨lder continuous of order less that p−2 4p . 2,2 3.2.13 Show that the process un,k deﬁned in (3.27) belongs to L(4) .

3.3 Anticipating stochastic diﬀerential equations The anticipating stochastic calculus developed in Section 3.2 can be applied to the study of stochastic diﬀerential equations where the solutions are nonadapted processes. Such kind of equations appear, for instance, when the initial condition is not independent of the Wiener process, or when we impose a condition relating the values of the process at the initial and ﬁnal times. In this section we discuss several simple examples of anticipating stochastic diﬀerential equations of Skorohod and Stratonovich types.

3.3.1 Stochastic diﬀerential equations in the Sratonovich sense In this section we will present examples of stochastic diﬀerential equations formulated using the Stratonovich integral. These equations can be solved taking into account that this integral satisﬁes the usual diﬀerentiation rules.

3.3 Anticipating stochastic diﬀerential equations

209

(A) Stochastic diﬀerential equations with random initial condition Suppose that W = {W (t), t ∈ [0, 1]} is a d-dimensional Brownian motion on the canonical probability space (Ω, F, P ). Let Ai : Rm → Rm , 0 ≤ i ≤ d, be continuous functions. Consider the stochastic diﬀerential equation Xt = X 0 +

d i=1

t

Ai (Xs ) ◦

dW Wsi

0

+

t

t ∈ [0, 1],

A0 (Xs )ds,

(3.49)

0

with an anticipating initial condition X0 ∈ L0 (Ω; Rm ). The stochastic integral is interpreted in the Stratonovich sense. There is a straightforward method to construct a solution to this equation, assuming that the coeﬃcients are smooth enough. The idea is to compose the stochastic ﬂow associated with the coeﬃcients with the random initial condition X0 . Suppose that the coeﬃcients Ai , 1 ≤ i ≤ d, are continuously

m d diﬀerentiable with bounded derivatives. Deﬁne B = A0 + 12 k=1 i=1 Aki ∂k Ai , and suppose moreover that B is Lipschitz. Let {ϕt (x), t ∈ [0, 1]} denote the stochastic ﬂow associated with the coeﬃcients of Eq. (3.49), that is, the solution to the following stochastic diﬀerential equation with initial value x ∈ Rm : ϕt (x)

= x+

d i=1

= x+

Ai (ϕs (x)) ◦

dW Wsi

t

+

A0 (ϕs (x))ds

0

d i=1

t

0

t

Ai (ϕs (x))dW Wsi +

t

B(ϕs (x))ds.

0

(3.50)

0

We know (see, for instance Kunita [173]) that there exists a version of ϕt (x) such that (t, x) → ϕt (x) is continuous, and we have p

p

E (|ϕt (x) − ϕs (y)| ) ≤ Cp,K (|t − s| 2 + |x − y|p ) for all s, t ∈ [0, 1], |x|, |y| ≤ K, p ≥ 2, K > 0. Then we can establish the following result. Theorem 3.3.1 Assume that Ai , 1 ≤ i ≤ d, are of class C 3 , and Ai , 1 ≤ i ≤ d, and B have bounded partial derivatives of ﬁrst order. Then for any random vector X0 , the process X = {ϕt (X0 ), t ∈ [0, 1]} satisﬁes the anticipating stochastic diﬀerential equation (3.49). Proof: Under the assumptions of the theorem we know that for any x ∈ Rm Eq. (3.50) has a unique solution ϕt (x). Set X = {ϕt (X0 ), t ∈ [0, 1]}. By a localization argument we can assume that the coeﬃcients Ai , 1 ≤ i ≤ d, and B have compact support. Then, it suﬃces to show that for any i = 1, . . . , d the process ui (t, x) = Ai (ϕt (x)) satisﬁes the hypotheses of

210

3. Anticipating stochastic calculus

Theorem 3.2.10. Condition (h1) is obvious and one easily check condition (h4) by means of Itˆ ˆo’s formula. This completes the proof. The uniqueness of a solution for Eq. (3.49) is more involved. We will 1,4 (Rm ), show that there is a unique solution in the class of processes L2,loc assuming some additional regulatiry conditions on the initial condition X0 . m Lemma 3.3.1 Suppose that X0 ∈ D1,p loc (R ) for some p > q ≥ 2, and assume that the coeﬃcients Ai , 1 ≤ i ≤ d, and B are of class C 2 with bounded partial derivatives of ﬁrst order. Then, we claim that {ϕt (X0 ), t ∈ m [0, 1]} belongs to L1,q 2,loc (R ).

Proof:

Let (Ωn , X0n ) be a localizing sequence for X0 in D1,p (Rm ). Set Gn,k,M = Ωn ∩ {|X0n | ≤ k} ∩ {

sup

|x|≤K,t∈[0,1]

|ϕt (x)| ≤ M }.

n n On the set Gn,k,M the process ϕt (X0 ) coincides with ϕM t (X0 β k (X0 )), where β k is a smooth function such that 0 ≤ β k ≤ 1, β k (x) = 1 if |x| ≤ k, and β k (x) = 0 if |x| ≥ k + 1, and ϕM t (x) is the stochastic ﬂow associated with the coeﬃcients β M Ai , 1 ≤ i ≤ d, and β M B. Hence, we can assume that the coeﬃcients Ai , 1 ≤ i ≤ d, and B are bounded and have bounded derivatives up to the second order, X0 ∈ D1,p (Rm ), and X0 is bounded by k. The following estimates hold

E

r

sup |x|≤K,t∈[0,1]

[|ϕt (x)|r + Dϕt (x)H ]

E

< ∞,

(3.51)

< ∞,

(3.52)

sup

|x|≤K,t∈[0,1]

|∇ϕt (x)|

q

for any r ≥ 2. The estimates follow from our assumptions on the coeﬃcients Ai , 1 ≤ i ≤ d, and B, taking into account that m d t ! " Dsj [ϕt (x)] = Aj (ϕs (x)) + ∂k Al (ϕr (x)Dsj ϕkr (x) dW Wrl +

k=1 l=1

m k=1

t

s

! " ∂k B(ϕr (x)Dsj ϕkr (x) dr,

(3.53)

s

for 1 ≤ j ≤ d and 0 ≤ s ≤ t ≤ 1. Hence, from Lemma 3.2.3 we obtain that {ϕt (X0 ), t ∈ [0, 1]} belongs to L1,q (Rm ) and Ds (ϕt (X0 )) = ∇ϕt (X0 )Ds X0 + (Dϕt ) (X0 ). Finally, from Eq. (3.53) for the derivative of ϕt (X0 ) it is easy to check that m {ϕt (X0 ), t ∈ [0, 1]} belongs to L1,q 2 (R ), and +,−

(D (ϕ(X0 )))t

+,−

= ∇ϕt (X0 )Dt X0 + (Dϕ)t

(X0 ).

3.3 Anticipating stochastic diﬀerential equations

211

Theorem 3.3.2 Assume that Ai , 1 ≤ i ≤ d, and B are of class C 4 , with m bounded partial derivatives of ﬁrst order and X0 belongs to D1,p loc (R ) for some p > 4. Then the process X = {ϕt (X0 ), t ∈ [0, 1]} is the unique solution 1,4 (Rm ) which is continuous. to Eq. (3.49) in L2,loc 1,4 (Rm ). Proof: By Lemma 3.3.1 we already know that X belongs to L2,loc n Let Y be another continuous solution in this space. Suppose that (Ω , X0n ) m n,1 , Y n ) is a localizing is a localizing sequence for X0 in D1,p loc (R ), and (Ω 1,4 sequence for Y in L2,loc (Rm ). Set

Ωn,k,M = Ωn ∩ Ωn,1 ∩ {|X0n | ≤ k} ∩ {

sup

|x|≤K,t∈[0,1]

|ϕt (x)| ≤ M }

∩{ sup |Y Yt | ≤ M }. t∈[0,1]

Ytn ), (β M Ai )(Y Ytn ), 0 ≤ i ≤ d, satisfy hypothesis (A1). The processes Ytn β M (Y n,k,M we have On the set Ω t d t i Yt = X0 + Ai (Y Ys ) ◦ dW Ws + A0 (Y Ys )ds, t ∈ [0, 1]. i=1

0

0

Let us denote by ϕ−1 t (x) the inverse of the mapping x → ϕt (x). By the classical Itˆo’s formula we can write d t −1 −1 (ϕs (ϕ−1 Ai (x) ◦ dW Wsi ϕt (x) = x − s (x)))

0

i=1 t

−1 (ϕs (ϕ−1 A0 (x)ds. s (x)))

− 0

i −1 −1 We claim that the processes Ft (x) = ϕ−1 Ai (x), t (x), Ht (x) = (ϕs (ϕs (x))) −1 −1 and Gt (x) = (ϕs (ϕs (x))) A0 (x) with values in the space of m × m matrices satisfy hypotheses (A2) to (A5). This follows from the properties of the stochastic ﬂow assicated with the coeﬃcients Ai . Consequently, we can apply Theorem 3.2.11 and we obtain, on the set Ωn,k,M d t −1 ϕs ϕ−1 (Y Y ) = X + (Y Ys )Ai (Y Ys ) ◦ dW Wsi t 0 t i=1

0

−1 ϕs (Y Ys )A0 (Y Ys )ds

t

+ 0

−

d i=1 t

t

(ϕs (ϕ−1 Ys )))−1 Ai (Y Ys ) ◦ dW Wsi s (Y

0

(ϕs (ϕ−1 Ys )))−1 A0 (Y Ys )ds. s (Y

− 0

212

3. Anticipating stochastic calculus

Notice that

−1 −1 (x) = −(ϕs (ϕ−1 . ϕs s (x)))

Hence, we get ϕ−1 Yt ) = X0 , that is, Yt = ϕt (X0 ) which completes the t (Y proof of the uniqueness. In [272] Ocone and Pardoux consider equations with a random drift and with time-dependent coeﬃcients. In such a case one has to remove the drift before composing with the random initial condition. The solution is given Yt ), where {ψ t (x), t ∈ [0, 1]} denotes the ﬂow associated with by Xt = ψ t (Y Eq. (3.49) with b ≡ 0, i.e., ψ t (x) = x +

d i=1

t

Ai (ψ s (x)) ◦ dW Wsi ,

0

and Yt solves the following ordinary diﬀerential equation parametrized by ω: Yt )dt, Y 0 = X0 , dY Yt = (ψ ∗t A0 )(Y −1 where (ψ ∗t A0 )(x) = ψ t (x) A0 (ψ t (x)). Then, a formal calculation based on Theorem 3.2.5 shows that the process Yt ), t ≥ 0} solves Eq. (3.49). In fact, we have {ψ t (Y d ψ t (Y Yt )

=

d

Ai (ψ t (Y Yt )) ◦ dW Wti + ψ t (Y Yt )Y Yt dt

i=1

=

d

Ai (ψ t (Y Yt )) ◦ dW Wti + A0 (ψ t (Y Yt ))dt.

i=1

(B) One-dimensional stochastic diﬀerential equations with random coeﬃcients Consider the one-dimensional equation t t Xt = X0 + σ(Xs ) ◦ dW Ws + b(Xs )ds, 0

(3.54)

0

where the coeﬃcients σ and b and initial condition X0 are random. Suppose that the coeﬃcients and the intial condition are deterministic and assume that σ is of class C 2 with bounded ﬁrst and second derivatives, and that b is Lipschtiz. Then, there is a unique solution Xt to Eq. (3.54) Wt , Yt ), where u(x, y) giben by (see Doss [84] and Exercise 2.2.2) Xt = u(W is the solution of the ordinary diﬀerential equation ∂u = σ(u), ∂x

u(0, y) = y,

3.3 Anticipating stochastic diﬀerential equations

213

and Yt is the solution to the ordinary diﬀerential equation

t

Yt = X0 +

f (W Ws , Ys )ds,

(3.55)

0

x where f (x, y) = b(u(x, y)) exp(− 0 σ (u(z, y)dz). Wt , Yt ) The next result shows that in the random case, the process Xt = u(W solves Eq. (3.54). Theorem 3.3.3 Suppose that the coeﬃcients and the intial condition of Eq. (3.54) are random and for each ω ∈ Ω, σ(ω) is of class C 2 with bounded Wt , Yt ), ﬁrst and second derivatives, and that b(ω) is Lipschtiz. Let Xt = u(W where Yt is given by (3.55). Then, σ(Xt ) is Stratonovich integrable on any interval [0, t] and (3.54) holds. Proof: Consider a partition π = {0 = t0 < t1 < · · · < tn = 1} of the interval [0, 1]. Denote by Wtπ the polygonal approximation of the Brownian Wtπ , Yt ). This process satisﬁes motion deﬁned in (3.5). Set Xtπ = u(W Xtπ

t

= X0 + = X0 +

0 Aπt

· σ(Xsπ )W s ds

+

t

Wsπ

exp

σ (u(y, Ys ))dy b(Xs )ds

0

Ws

+ Btπ .

The term Aπt can be written as Aπt

t

[σ(Xsπ )

= 0

·

t

·

σ(Xs )W s ds = A1,π + A2,π t t .

− σ(Xs )] W s ds + 0

t Clearly Aπt = Xtπ −X0 −Btπ converges in probability to Xt −X0 − 0 b(Xs )ds can be expressed as Riemann as |π| tends to zero. On the other hand, A2,π t t Ws . As a sum that approximates the Stratonovich integral 0 σ(Xs ) ◦ dW 1,π consequence, it suﬃces to show that At converges in probability to zero as |π| tends to zero. We have σ(Xsπ ) − σ(Xs ) = σσ (Xs ) (W Wsπ − Ws ) 1 2 Wsπ − Ws ) , σ(σ )2 + σ 2 σ (ξ) (W + 2 where ξ is a random intermediate point between Wsπ and Ws . The last summand of the above expression does not contribute tot the limit of A1,π t . In order to handle the ﬁrst summand we set Zs = σσ (Xs ). To simplify we assume t = 1. Consider another partition of the interval [0, 1] of the form

214

{sj =

3. Anticipating stochastic calculus

≤ j ≤ m}. We can write m−1 1 sj+1 · · π π ≤ Z (W W − W ) W ds Z (W W − W ) W ds s s s sj s s s s 0 sj j=1 m−1 · sj+1 + Zs − Zsj (W Wsπ − Ws ) W s ds j=1 sj j m, 0

= C π,m + Dπ,m . For any ﬁxed m the term C π,m converges in probability to zero as |π| tends to zero. In fact, if the points sj and sj+1 belong to the partition π, we have sj+1 · (W Wsπ − Ws ) W s ds sj

=

i:ti ∈[sj ,sj+1 )

= − ×

Wti+1 − Wti ti+1 − ti

i:ti ∈[sj ,sj+1 )

ti+1

ti

ti+1

(W Wsπ − Ws ) ds

ti

1 2(ti+1 − ti )

Wti+1 − Ws

2

− (W Ws − Wti )2 ds

and this converges to zero. Finally, the term Dπ,m can be estimated as follows m−1 sj+1 · π (W W sup |Zs − Zt | − W ) W |Dπ,m | ≤ s s ds s 1 |t−s|≤ m

≤

sup 1 |t−s|≤ m

×

ti+1

ti

j=1

|Zs − Zt |

m−1

sj

j=0 i:ti ∈[sj ,sj+1 )

1 2(ti+1 − ti )

2 Wti+1 − Ws + (W Ws − Wti )2 ds.

As |π| tends to zero the right-hand side of the above inequality converges in probability to 12 . Hence, we obtain lim sup |Dπ,m | ≤ |π|→0

1 sup |Zs − Zt | , 2 |t−s|≤ 1 m

and this expression tends to zero in probability as m tends to inﬁnity, due to the continuity of the process Zt .

3.3 Anticipating stochastic diﬀerential equations

215

3.3.2 Stochastic diﬀerential equations with boundary conditions Consider the following Stratonovich diﬀerential equation on the time interval [0, 1], where instead of giving the value X0 , we impose a linear relationship between the values of X0 and X1 : 2

d dXt = i=1 Ai (Xt ) ◦ dW Wti + A0 (Xt )dt, (3.56) h(X0 , X1 ) = 0. We are interested in proving the existence and uniqueness of a solution for this type of equations. We will discuss two particular cases: (a) The case where the coeﬃcients Ai , 0 ≤ i ≤ d and the function h are aﬃne (see Ocone and Pardoux [273]). (b) The one-dimensional cased (see Donati-Martin [80]). (A) Linear stochastic diﬀerential equations with boundary conditions Consider the following stochastic boundary value problem for t ∈ [0, 1] 2 t

d t Wsi + 0 A0 Xs ds, Xt = X0 + i=1 0 Ai Xs ◦ dW (3.57) H0 X0 + H1 X1 = h. We assume that Ai , i = 0, . . . , d, H0 , and H1 are m × m deterministic H0 : H 1 ) matrices and h ∈ Rm . We will also assume that the m×2m matrix (H has rank m. Concerning the boundary condition, two particular cases are interesting: Two-point boundary-value problem: Let l∈ N be such that 0 < l < m. H0 0 Suppose that H0 = , H1 = , where H0 is an l × m matrix 0 H1 and H1 is an (m − l) × m matrix. Condition rank(H H0 : H1 ) = mimplies h0 that H0 has rank l and that H1 has rank m − l. If we write h = , h1 where h0 ∈ Rl and h1 ∈ Rm−l , then the boundary condition becomes H0 X0 = h0 ,

H1 X1 = h1 .

Periodic solution: Suppose H0 = −H1 = I and h = 0. Then the boundary condition becomes X0 = X1 . With (3.57) we can associate an m × m adapted and continuous matrixvalued process Φ solution of the Stratonovich stochastic diﬀerential equation

216

3. Anticipating stochastic calculus

2

dΦt =

d i=1

Ai Φt ◦ dW Wti + BΦt dt,

(3.58)

Φ0 = I. Using the Itˆ oˆ formula for the Stratonovich integral, one can obtain an explicit expression for the solution to Eq. (3.57). By a solution we mean a continuous and adapted stochastic process X such that Ai (Xt ) is Stratonovich integrable with respect to W i on any time interval [0, t] and such that (3.57) holds. Theorem 3.3.4 Suppose that the random matrix H0 + H1 Φ1 is a.s. invertible. Then there exists a solution to the stochastic diﬀerential equation (3.57) which is unique among those continuous processes whose compo2,4 . nents belong to the space LS,loc Proof:

Deﬁne Xt = Φt X0 ,

(3.59)

where X0 is given by −1

X0 = [H H0 + H1 Φ1 ]

h.

(3.60)

2,4 Then it follows from this expression that X i belongs to LS,loc , for all i = 1, . . . , m, and due to the change-of-variables formula for the Stratonovich integral (Theorem 3.2.6), this process satisﬁes Eq. (3.57). In order to show the uniqueness, we proceed as follows. Let Y ∈ 2,4 (Rm ) be a solution to (3.57). Then we have LS,loc

Φ−1 t =I −

d i=1

t

Φ−1 Wsi − s Ai ◦ dW

0

t

Φ−1 s A0 ds.

0

By the change-of-variables formula for the Stratonovich integral (Theorem 3.2.6) we see that Φ−1 t Yt , namely, Yt = Φt Y0 .Therefore, Y satisﬁes (3.59). But (3.60) follows from (3.59) and the boundary condition H0 Y0 + H1 Y1 = h. Consequently, Y satisﬁes (3.59) and (3.60), and it must be equal to X. 2,4 m Notice that, in general, the solution to (3.57) will not belong to LS,loc (R ). One can also treat non-homogeneous equations of the form 2 t

d t Xt = X0 + i=1 0 Ai Xs ◦ dW Wsi + 0 A0 Xs ds + Vt , (3.61) H0 X0 + H1 X1 = h, where Vt is a continuous semimartingale. In that case, & ' 1 t −1 Xt = Φt [H H0 + H1 Φ1 ] Φ−1 ◦ dV V Φ−1 Vs h − H1 Φ1 + Φ s t s s ◦ dV 0

0 2,4 LS,loc (Rm )

is a solution to Eq. (3.61). The uniqueness in the class established provided the process Vt also belongs to this space.

can be

3.3 Anticipating stochastic diﬀerential equations

217

(B) One-dimensional stochastic diﬀerential equations with boundary conditions Consider the one-dimensional stochastic boundary value problem 2 t t Xt = X0 + 0 σ(Xs ) ◦ dW Ws + 0 b(Xs )ds,

(3.62)

a0 X0 + a1 X1 = a2 , Applying the techniques of the anticipating stochastic calculus we can show the following result. Theorem 3.3.5 Suppose that the functions σ and b1 := b + 12 σσ are of class C 2 with bounded derivatives and a0 a1 > 0. Then there exists a solution to Eq. (3.62). Furthermore, if the functions σ and b1 are of class C 4 with 2,4 . bounded derivatives then the solution is unique in the class LS,loc Proof: Let ϕt (x) be the stochastic ﬂow associated with the coeﬃcients σ and b1 . By Theoren 3.3.1 for any random variable X0 the process ϕt (X0 ) satisﬁes t

t

σ(Xs ) ◦ dW Ws +

Xt = X0 +

b(Xs )ds.

0

0

Hence, in order to show the existence of a solution it suﬃces to prove that there is a unique random variable X0 such that ϕ1 (X0 ) =

a2 − a0 X0 . a1

(3.63)

The mapping g(x) = ϕ1 (x) is strictly increasing and this implies the existence of a unique solution to Eq. (3.63). Taking into account Theorem 3.3.2 to show the uniqueness it suﬃces to check that the unique solution X0 to Eq. (3.63) belongs to D1,p loc for some p > 4. By the results of Doss (see [84] and Exercise 2.2.2) one can represent the ﬂow ϕt (x) as a Fr´´echet diﬀerentiable function of the Brownian motion W . Using this fact and the implicit function theorem one deduces that X0 ∈ D1,p loc for all p ≥ 2.

3.3.3 Stochastic diﬀerential equations in the Skorohod sense Let W = {W Wt , t ∈ [0, 1]} be a one-dimensional Brownian motion deﬁned on the canonical probability space (Ω, F, P ). Consider the stochastic diﬀerential equation t t σ(s, Xs )dW Ws + b(s, Xs )ds, (3.64) Xt = X0 + 0

0

0 ≤ t ≤ 1, where X0 is F1 -measurable and σ and b are deterministic functions. First notice that the usual Picard iteration procedure cannot

218

3. Anticipating stochastic calculus

be applied in that situation. In fact the estimation of the L2 -norm of the Skorohod integral requires a bound for the derivative of X, and this derivative can be estimated only in terms of the second derivative. So we are faced with a nonclosed procedure. In some sense, Eq. (3.64) is an inﬁnite dimensional hyperbolic partial diﬀerential equation. In fact, this equation can be formally written as

t

1 σ(s, Xs ) ◦ dW Ws + 2

Xt = X 0 + 0 t + b(s, Xs )ds.

t

σ (s, Xs )

0

! + " D X s + D− X s ds

0

Nevertheless, if the diﬀusion coeﬃcient is linear it is possible to show that there is a unique global solution using the techniques developed by Buckdahn in [48, 49], based on the classical Girsanov transformation. In order to illustrate this approcah let consider the following particular case: Xt = X0 + σ

t

Xs dW Ws .

(3.65)

0

When X0 is deterministic, the solution to this equation is the martingale 2

1

Wt − 2 σ t Xt = X0 eσW .

If X0 = signW W1 , then a solution to Eq. (3.65) is (see Exercise 3.2.3) 1

2

Wt − 2 σ t Xt = sign (W W1 − σt) eσW .

More generally, by Theorem 3.3.6 below, if X0 ∈ Lp (Ω) for some p > 2, then Wt − 12 σ 2 t Xt = X0 (At )eσW is a solution to Eq. (3.65), where At (ω)s = ω s − σ(t ∧ s). In terms of the Wick product (see [53]) one can write 1

2

Wt − 2 σ t . Xt = X0 ♦eσW

Let us now turn to the case of a general linear diﬀusion coeﬃcient and consider the equation Xt = X0 +

t

σ s Xs dW Ws + 0

t

b(s, Xs )ds,

0 ≤ t ≤ 1,

(3.66)

0

where σ ∈ L2 ([0, 1]), X0 is a random variable and b is a random function satisfying the following condition:

3.3 Anticipating stochastic diﬀerential equations

219

(H.1) b : [0, 1] × R × Ω → R is a measurable function such that there exist an integrable function γ t on [0, 1], γ t ≥ 0, a constant L > 0 , and a set N1 ∈ F of probability one, verifying 1 γ t dt ≤ L, |b(t, x, ω) − b(t, y, ω)| ≤ γ t |x − y| , |b(t, 0, ω)|

0

≤ L,

for all x, y ∈ R, t ∈ [0, 1] and ω ∈ N1 . Let us introduce some notation. Consider the family of transformations Tt , At : Ω → Ω, t ∈ [0, 1] , given by

t∧s

Tt (ω)s = ω s +

σ u du, 0

t∧s

At (ω)s = ω s −

σ u du . 0

Note that Tt At = At Tt = Identity. Deﬁne t 1 t 2 σ s dW Ws − σ s ds . εt = exp 2 0 0 Then, by Girsanov’s theorem (see Proposition 4.1.2) E [F (At )εt ] = E[F ] for any random variable F ∈ L1 (Ω). For each x ∈ R and ω ∈ Ω we denote by Zt (ω, x) the solution of the integral equation Zt (ω, x) = x +

t

ε−1 Tt (ω)) b (s, εs (T Tt (ω)) Zs (ω, x), Ts (ω)) ds . (3.67) s (T

0

s s Notice that for s ≤ t we have εs (T Tt ) = exp 0 σ u dW Wu + 12 0 σ 2u du = Ts ). Henceforth we will omit the dependence on ω in order to simplify εs (T the notation. Theorem 3.3.6 Fix an initial condition X0 ∈ Lp (Ω) for some p > 2, and deﬁne (3.68) Xt = εt Zt (At , X0 (At )) . Then the process X = {Xt , 0 ≤ t ≤ 1} satisﬁes 1[0,t] σX ∈ Dom δ for all t ∈ [0, 1], X ∈ L2 ([0, 1] × Ω), and X is the unique solution of Eq. (3.66) verifying these conditions. Proof: Existence: Let us prove ﬁrst that the process X given by (3.68) satisﬁes the desired conditions. By Gronwall’s lemma and using hypothesis (H.1), we have t ε−1 (T T )ds , |Xt | ≤ εt etL |X0 (At )| + L s s 0

220

3. Anticipating stochastic calculus

which implies supt∈[0,1] E(|Xt |q ) < ∞, for all 2 ≤ q < p, as it follows easily from Girsanov’s theorem and H¨ older’s inequality. Indeed, we have t q qtL q q q −q E(|Xt | ) ≤ cq E εt e εs (T Ts )ds |X0 (At )| + L 0 t q−1 qL q q q−1 −q 2 ≤ cq e Tt )|X0 | + L εt (T Tt ) εs (T Ts )ds E εt (T 0 4 5 q ≤ C E (|X0 |p ) p + 1 . Now ﬁx t ∈ [0, 1] and let us prove that 1[0,t] σX ∈ Dom δ and that (3.66) holds. Let G ∈ S be a smooth random variable. Using (3.68) and Girsanov’s theorem, we obtain t t E σ s Xs Ds Gds = E σ s εs Zs (As , X0 (As )) Ds Gds 0 0 t = E σ s Zs (X0 ) (Ds G) (T Ts )ds . (3.69) 0 d G(T Ts ) = σ s (Ds G) (T Ts ). Therefore, integrating by parts in Notice that ds (3.69) and again applying Girsanov’s theorem yield t d E Ts )dss = E Zt (X0 )G(T Zs (X0 ) G(T Tt ) − Z0 (X0 )G ds 0 t − ε−1 (T T )b (s, ε (T T )Z (X ), T ) G(T T )ds t s s s 0 s s s 0

=E (εt Zt (At , X0 (At )) G) − E (Z Z0 (X0 )G) t E (b (s, εs Zs (As , X0 (As ))) G) ds − 0 t E (b(s, Xs )G) ds . =E (Xt G) − E (X0 G) − 0

t

Because the random variable Xt − X0 − 0 b(s, Xs )ds is square integrable, we deduce that 1[0,t] σX belongs to the domain of δ and that (3.66) holds. Uniqueness: Let Y be a solution to Eq. (3.66) such that Y belongs to L2 ([0, 1] × Ω) and 1[0,t] σY ∈ Dom δ for all t ∈ [0, 1]. Fix t ∈ [0, 1] and let G be a smooth random variable. Multiplying both members of (3.66) by G(At ) and taking expectations yield t E (Y Yt G(At )) = E (Y Y0 G(At )) + E b(s, Ys )G(At )ds 0 t +E σ s Ys Ds (G(At )) ds . (3.70) 0

3.3 Anticipating stochastic diﬀerential equations

Notice that obtains

d ds G(As )

221

= −σ(Ds G)(As ). Therefore, integrating by parts

t Y0 G) − E (Y Y0 σ s (Ds G)(As )) ds E (Y Yt G(At )) = E (Y 0 t t r b(s, Ys )G(As )ds − E b(s, Ys )σ r (Dr G)(Ar )dsdr +E 0 0 0 t +E σ s Ys (Ds G)(As )ds 0 t r −E σ s Ys (Dr Ds G)(Ar )σ r dsdr . (3.71) 0

0

If we apply Eq. (3.71) to the smooth random variable σ r (Dr G)(Ar ) for each ﬁxed r ∈ [0,t], the negative terms in the above expression cancel out t with the term E 0 σ s Ys (Ds G) (As )ds , and we obtain

t

E (Y Yt G(At )) = E (Y Y0 G) + E

b(s, Ys )G(As ) ds .

0

By Girsanov’s theorem this implies E

Yt (T Tt )ε−1 Tt )G t (T

= E (Y Y0 G) + E

t

b (s, Ys (T Ts ), Ts ) ε−1 Ts )G s (T

ds.

0

Therefore, we have Yt (T Tt )ε−1 Tt ) = Y0 + t (T

t

b (s, Ys (T Ts ), Ts ) ε−1 (T Ts )ds,

(3.72)

0

and from (3.72) we get Yt (T Tt )ε−1 Tt ) = Zt (Y Y0 ) a.s. That is, t (T Yt = εt Zt At , Y0 (At ) = Xt a.s., which completes the proof of the uniqueness.

When the diﬀusion coeﬃcient is not linear one can show that there exists a solution up to a random time.

Exercises 3.3.1 Let f be a continuously diﬀerentiable function with bounded derivative. Solve the linear Skorohod stochastic diﬀerential equation dXt

= Xt dW Wt ,

X0

= f (W W1 ).

t ∈ [0, 1]

222

3. Anticipating stochastic calculus

3.3.2 Consider the stochastic boundary-value problem dXt1 dXt2 X01

= Xt2 ◦ dW Wt , = 0, =

0,

X11 = 1.

Find the unique solution of this system, and show that E(|Xt |2 ) = ∞ for all t ∈ [0, 1]. 3.3.3 Find an explicit solution for the stochastic boundary-value problem dXt1 dXt2 X01 + X02

=

(Xt1 + Xt2 ) ◦ dW Wt1 ,

= Xt2 ◦ dW Wt2 , = 1, X12 = 1.

Notes and comments [3.1] The Skorohod integral is an extension of the Itˆ oˆ integral introduced in Section 1.3 as the adjoint of the derivative operator. In Section 3.1, following [249], we show that it can be obtained as the limit of two types of modiﬁed Riemann sums, including the conditional expectation operator or subtracting a complementary term that converges to the trace of the derivative. The forward stochastic integral, deﬁned as δ(u) +

1

Dt− ut dt,

0

is also an extension of the Ito ˆ integral which has been studied by diﬀerent authors. Berger and Mizel [30] introduced this integral in order to solve stochastic Volterra equations. In [14], Asch and Potthoﬀ prove that it satisﬁes a change-of-variables formula analogous to that of the Itˆ oˆ calculus. An approach using a convolution of the Brownian path with a rectangular function can be found in Russo and Vallois [298]. In [176] Kuo and Russek study the anticipating stochastic integrals in the framework of the white noise calculus. The deﬁnition of the stochastic integral using an orthonormal basis of L2 ([0, 1]) is due to Paley and Wiener in the case of deterministic integrands. For random integrands this analytic approach has been studied by Balkan [16], Ogawa [274, 275, 276], Kuo and Russek [176] and Rosinski [293], among others. [3.2] The stochastic calculus for the Skorohod and Stratonovich integrals was developed by Nualart and Pardoux [249]. In particular, the local

3.3 Anticipating stochastic diﬀerential equations

223

property introduced there has allowed us to extend the change-of-variables formula and to deal with processes that are only locally integrable or possess locally integrable derivatives. Another extensive work on the stochastic calculus for the Skorohod integral in L2 is Sekiguchi and Shiota [305]. Other versions of the change-of-variables formula for the Skorohod inte¨ ¨ gral can be found in Sevljakov [306], Hitsuda [136], and Ustunel [330]. [3.3] For a survey of this kind of applications, we refer the reader to Pardoux [278]. Stochastic diﬀerential equations in the Skorohod sense were ﬁrst studied by Shiota [311] using Wiener chaos expansions. A diﬀerent method was ¨ ¨ [331]. The approach described in Section 3.3, based on the used by Ustunel Girsanov transformation, is due to Buckdahn and allows us to solve a wide class of quasilinear stochastic diﬀerential equations in the Skorohod sense with a constant or adapted diﬀusion coeﬃcient. When the diﬀusion coeﬃcient is random, one can use the same ideas by applying the anticipating version of Girsanov’s theorem (see [50]). In [49] Buckdahn considers Skorohod stochastic diﬀerential equations of the form (3.64), where the diﬀusion coeﬃcient σ is not necessarily linear. In this case the situation is much more complicated, and an existence theorem can be proved only in some random neighborhood of zero. Stochastic diﬀerential equations in the sense of Stratonovich have been studied by Ocone and Pardoux in [272]. In this paper they prove the existence of a unique solution to Eq. (3.49) assuming that the coeﬃcient A0 (s, x) is random. In [273] Ocone and Pardoux treat stochastic diﬀerential equations of the Stratonovich type with boundary conditions of the form (3.56), assuming that the coeﬃcients Ai and the function h are aﬃne, and they also investigate the Markov properties of the solution.

4 Transformations of the Wiener measure

In this chapter we discuss diﬀerent extensions of the classical Girsanov theorem to the case of a transformation of the Brownian motion induced by a nonadapted process. This generalized version of Girsanov’s theorem will be applied to study the Markov property of solutions to stochastic diﬀerential equations with boundary conditions.

4.1 Anticipating Girsanov theorems In this section we will work in the context of an abstract Wiener space. That is, we will assume that the underlying probability space (Ω, F, P ) is such that Ω is a separable Banach space, P is a Gaussian measure on Ω with full support, and F is the completion of the Borel σ-ﬁeld of Ω with respect to P . Moreover, there is a separable Hilbert space H that is continuously and densely embedded in Ω, with injection i : H → Ω, and such that 1 eix,y P (dx) = exp(− y2H ) 2 Ω for any y ∈ Ω∗ ⊂ H (here we identify H with its dual). The triple (Ω, H, P ) is called an abstract Wiener space. Note that each element y ∈ Ω∗ deﬁnes a Gaussian random variable. If we denote this variable by W (y), then the mapping y → W (y), from Ω∗ into L2 (Ω), is continuous with respect to the norm of H, and it can be extended to H. In that way H is isometric to a Gaussian subspace in L2 (Ω) that generates F

226

4. Transformations of the Wiener measure

and that is denoted by H1 (i.e., H1 is the ﬁrst Wiener chaos). The image of H by the injection i is denoted by H 1 ⊂ Ω. The classical Wiener space is a particular case of an abstract Wiener t space if we take Ω = C0 ([0, 1]), H = L2 ([0, 1]), i(h)(t) = 0 h(s)ds, and P is the Wiener measure. The space H 1 is here the Cameron-Martin space. Consider a measurable mapping u : Ω → H, and deﬁne the transformation T : Ω → Ω by T (ω) = ω + i(u(ω)). (4.1) In the white noise case, H can be represented as H = L2 (T, B, µ), and Hvalued random variables are stochastic processes parametrized by T . Along this chapter we will use this terminology. We want to discuss the following problems: (A) When is the image measure P ◦ T −1 absolutely continuous with respect to P ? (B) Find a new probability Q absolutely continuous with respect to P such that Q ◦ T −1 = P . Furthermore, we are also interested in ﬁnding expressions for the Radon◦T −1 ] Nikodym densities d[P dP and dQ dP in terms of u.

4.1.1

The adapted case

Consider the case of the Wiener space, that is, Ω = C0 ([0, 1]). Then u = 1 {ut , 0 ≤ t ≤ 1} is a random process such that 0 u2t dt < ∞ a.s., and the associated transformation is given by t T (ω)t = ω t + us (ω)ds. (4.2) 0

Suppose that the process u is adapted, and deﬁne t 1 t 2 ξ t = exp − us dW Ws − u ds , t ∈ [0, 1]. 2 0 s 0

(4.3)

The following two propositions provide a complete answer to questions (A) and (B). Proposition 4.1.1 The probability P ◦ T −1 is absolutely continuous with respect to P . Proposition 4.1.2 (Girsanov theorem) There exists a probability Q absolutely continuous with respect to P such that Q ◦ T −1 = P (that is, t Wt + 0 us ds has the law of a Brownian motion under Q) if and only if E(ξ 1 ) = 1 and in this case dQ dP = ξ 1 .

4.1 Anticipating Girsanov theorems

227

Proof of Proposition 4.1.2: Suppose ﬁrst that E(ξ 1 ) = 1. Consider the increasing family of stopping times deﬁned by

t

u2s ds ≥ k}.

τ k = inf{t : 0

By Itˆˆo’s formula {ξ t∧τ k , t ∈ [0, 1]} is a positive continuous martingale. Hence {ξ t , t ∈ [0, 1]} is a continuous local martingale, and is a martingale because E(ξ 1 ) = 1. Fix 0 ≤ s < t ≤ 1 and A ∈ Fs . Applying Itˆ oˆ’s formula yields Wt −W Ws + st ur dr) E eiλ(W 1A ξ 1 Wt −W Ws + st ur dr)− st ur dW Wr − 12 st u2r dr |F Fs 1A ξ s = E E eiλ(W t 2 λ Wu −W Ws + su uθ dθ) E eiλ(W 1A ξ 1 du. = E(1A ξ 1 ) − s 2 Hence,

λ2 Wt −W Ws + st ur dr) E eiλ(W 1A ξ 1 = E(1A ξ 1 )e− 2 (t−s) ,

and we obtain E(F (T )ξ 1 ) = E(F ) for any functional F : Ω → C of the form ⎛ ⎞ m−1 F = exp⎝i λj (W (tj+1 ) − W (tj )⎠ , j=0

where λj ∈ R and 0 = t0 < t1 < · · · < tm = 1. Therefore, the probability −1 Q given by dQ = P. dP = ξ 1 satisﬁes Q ◦ T −1 Conversely, assume Q ◦ T = P and dQ dP = η. For any integer k ≥ 1 we deﬁne the transformation t∧τ k us ds. Tk (ω)t = ω t + 0

In view of E(ξ τ k ) = 1 we can apply the arguments of the ﬁrst part of the proof to the transformation Tk , deducing E(F (T Tk )ξ τ k ) = E(F ) for any nonnegative and bounded random variable F . If F is Fτ k -measurable, then F (T Tk ) = F (T ). On the other hand, we know that E(F (T )η) = E(F ).

228

4. Transformations of the Wiener measure

Hence, ξ τ k = E(η|T −1 (F Fτ k )), and letting k tend to inﬁnity obtains ξ 1 = η. Proof of Proposition 4.1.1: Let B be a Borel subset of Ω with P (B) = 0. Consider the stopping times τ k and the transformations Tk introduced in the proof of Proposition 4.1.2. We know that P ◦ Tk−1 and P are mutually absolutely continuous. Hence, P (T −1 (B)) = P (T −1 (B) ∩ {τ k = 1}) + P (T −1 (B) ∩ {τ k < 1}) = P (T Tk−1 (B) ∩ {τ k = 1}) + P (T −1 (B) ∩ {τ k < 1}) 1 u2t dt > k , ≤ P {τ k < 1} = P 0

which converges to zero as k tends to inﬁnity. This completes the proof.

4.1.2 General results on absolute continuity of transformations In this subsection we will assume that (Ω, F, P ) is an arbitrary complete probability space and T : Ω −→ Ω is a measurable transformation. First we will see that questions (A) and (B) are equivalent if the measures P ◦ T −1 and P (or Q and P ) are equivalent (i.e., they are mutually absolutely continuous). Lemma 4.1.1 The following statements are equivalent: (i) The probabilities P ◦ T −1 and P are equivalent. (ii) There exists a probability Q equivalent to P such that Q ◦ T −1 = P . −1

◦T ] and Y = dQ Under the above assumptions, and setting X = d[P dP dP , we 1 have E(Y | T ) = X(T ) , P a.s. If we assume, moreover, that {T −1 (A), A ∈ F} = F a.s. (which is equivalent to the existence of a left inverse Tl such that Tl−1 ◦ T = I a.s.), we have

dQ = dP

d[P ◦ T −1 ] ◦T dP

−1 .

(4.4) −1

◦T ] . We know Proof: Let us ﬁrst show that (i) implies (ii). Set X = d[P dP −1 that P {X = 0} = 0 and P {X(T ) = 0} = (P ◦ T ){X = 0} = 0. Deﬁne 1 the measure Q by dQ = X(T ) dP . Then for any B ∈ F we have 1 1 −1 dP = d[P ◦ T −1 ] = P (B). Q(T (B)) = X(T ) X T −1 (B) B

Clearly, P << Q because

dQ dP

= 0.

4.1 Anticipating Girsanov theorems

229

Conversely, assume that (ii) holds. Then (i) follows from the implications P (B) = 0

⇐⇒

Q(T −1 (B)) = 0

⇐⇒

P (T −1 (B)) = 0.

1 In order to show the equality E(Y | T ) = X(T ) a.s., we write for any B∈F E(Y |T )dP = Y dP = Q(T −1 (B)) T −1 (B) T −1 (B) 1 = dP. T −1 (B) X(T )

The last statement follows from E(Y | T ) = Y .

Remarks: If we only assume the absolute continuity P ◦ T −1 << P (or Q << P ), the above equivalence is no longer true. We can only aﬃrm that Q ◦ T −1 = P and P << Q imply P ◦ T −1 << P . The following examples illustrate this point. Examples: Let Ω = C0 ([0, 1]) and P be the Wiener measure. Consider the following two examples. (1) Suppose ut = f (W W1 ), where f : R → R is a measurable function. The probability P ◦T −1 is absolutely continuous with respect to P if and only if the distribution of the random variable Y + f (Y ) is absolutely continuous, where Y has the law N (0, 1) (see Exercise 4.1.1). If we take f (x) = x2 , then P ◦ T −1 << P , but it is not possible to ﬁnd a probability Q on Ω such W12 , t ∈ [0, 1]} is a Brownian motion. that under Q the process {W Wt + tW (2)

Let ut =

2W Wt 1[0,S] (t), (1 − t)2

where S = inf{t : Wt2 = 1 − t}. Since P (0 < S < 1) = 1, we have 1 2 S Wt2 u dt = 4 0 (1−t) 4 dt < ∞ a.s. However, E(ξ 1 ) < 1 (see Exercise 4.1.2), 0 t where ξ 1 is given by (4.3), and, from Proposition 4.1.2, it is not possible to ﬁnd a probability Q absolutely continuous with respect to P such that under Q the process t S∧t Ws us ds = Wt + 2 ds Wt + (1 − s)2 0 0 is a Brownian motion (see [200, p. 224]). In the following sections we will look for a random variable η such that E(η1T −1 (B) ) ≤ P (B)

(4.5) −1

for all B ∈ F. If η > 0 a.s., this implies that P ◦ T << P . On the other hand, if the equality in (4.5) holds for all B ∈ F (which is equivalent to imposing that E(η) = 1), then P is equivalent to P ◦ T −1 , and the −1 = P. probability Q given by dQ dP = η veriﬁes Q ◦ T

230

4. Transformations of the Wiener measure

4.1.3 Continuously diﬀerentiable variables in the direction of H 1 Let (Ω, H, P ) be an abstract Wiener space. We will need the following notion of diﬀerentiability. Deﬁnition 4.1.1 We will say that a random variable F is (a.s.) H-continuously diﬀerentiable if for (almost) all ω ∈ Ω the mapping h → F (ω + i(h)) is continuously diﬀerentiable in H. The set of a.s. H-continuously 1 . diﬀerentiable functions will be denoted by CH 1,2 1 We will prove that CH ⊂ Dloc . In order to show this inclusion we have to introduce some preliminary tools. For any set A ∈ F deﬁne

ρA (ω) = inf{hH : ω + i(h) ∈ A}, where the inﬁmum is taken to be inﬁnite if the set is empty. Clearly, ρA (ω) = 0 if ω ∈ A, and ρA (ω) ≤ if ω belongs to the -neighborhood of A constructed with the distance of H 1 . Also, ρA (ω) = ∞ if ω ∈ A + H 1 . Other properties of this distance are the following: (i) A ⊂ B

= =⇒

ρA (ω) ≥ ρB (ω);

(ii) |ρA (ω + i(h)) − ρA (ω)| ≤ hH ; (iii) An ↑ A

= =⇒

ρAn (ω) ↓ ρA (ω);

(iv) If G is σ-compact and φ ∈ C0∞ (R), then φ(ρG ) belongs to D1,p for all p ≥ 2 (with the convention φ(∞) = 0), and D[φ(ρG )]H ≤ φ ∞ . Proof of these properties: Properties (i) and (ii) are immediate consequences of the deﬁnition of ρA . In order to show (iii) it suﬃces to see that inf n ρAn (ω) ≤ ρA (ω) for all ω ∈ Ω. If ω ∈ A + H 1 , this inequality is clear. Suppose ω ∈ A + H 1 . By deﬁnition, for any > 0 there exists h ∈ H such that ω + i(h) ∈ A and hH ≤ ρA (ω) + . We can ﬁnd an index n0 such that ω + i(h) ∈ An for all n ≥ n0 . Therefore, ρAn (ω) ≤ ρA (ω) + for all n ≥ n0 , which implies the desired result. Kn } is an increasing sequence Let us prove (iv). Set G = ∪n Kn , where {K of compact sets. By (iii) we have ρKn ↓ ρG , hence φ(ρKn ) → φ(ρG ). For each n, ρKn is a measurable function because it can be written as ∞ on (K Kn + H 1 )c ρKn (ω) = dH 1 ((ω − Kn ) ∩ H 1 , 0) on Kn + H 1 , where dH 1 is the metric of H 1 and ω → (ω − Kn ) ∩ H 1 is a measurable function from Kn + H 1 into the closed subsets of H 1 . Consequently, φ(ρG ) is a bounded and measurable function such that |φ(ρG (ω + i(h))) − φ(ρG (ω))| ≤ φ ∞ hH , and it suﬃces to apply Exercise 1.2.9 and Proposition 1.5.5.

(4.6)

4.1 Anticipating Girsanov theorems

231

1,2 1 Proposition 4.1.3 We have CH ⊂ Dloc .

Proof:

1 Suppose that F ∈ CH . Deﬁne the following sequence of sets:

4 An = ω ∈ Ω :

sup 1

h H ≤ n

sup 1

h H ≤ n

|F (ω + i(h))| ≤ n, 5 DF (ω + i(h))H ≤ n .

The fact ;∞that F is a.s. H-continuously diﬀerentiable implies that almost surely n=1 An = Ω. For each n we can ﬁnd a σ-compact set Gn ⊂ An such that P (Gn ) = P (An ). Let φ ∈ C0∞ (R) be a nonnegative function such that |φ(t)| ≤ 1 and |φ (t)| ≤ 4 for all t, φ(t) = 1 for |t| ≤ 13 , and φ(t) = 0 for |t| ≥ 23 . Deﬁne Fn = φ(nρGn )F. We have (a) Fn = F on the set Gn . (b) Fn is bounded by n. Indeed, |F Fn | ≤ 1{ρG

n

2 < 3n } |F |

≤ n,

2 implies that there exists h ∈ H with ω + i(h) ∈ because ρGn < 3n 2 ≤ n1 . Gn ⊂ An and hH < 3n

(c) For any k ∈ H with kH ≤

1 3n

we have, using (4.6),

|F Fn (ω + i(k)) − Fn (ω)| ≤ |φ(nρGn (ω + i(k)))F (ω + i(k)) − φ(nρGn (ω))F (ω + i(k))| + |φ(nρGn (ω))F (ω + i(k)) − φ(nρGn (ω))F (ω)| 2 2 ≤ 4nkH 1{ρG < 3n or ρG −i(k) < 3n } |F (ω + i(k))| n

n

|F (ω + i(k)) − F (ω)| 1 2 2 ≤ 8n kH + 1{ρG < 3n } kH DF (ω + ti(k))H dt + 1{ρG

2 < 3n } n

n

0

≤ (8n + n)kH . 2

So, by Exercise 1.2.9 we obtain Fn ∈ D1,2 , and the proof is complete. 1 Lemma 4.1.2 Let F be a random variable in CH such that E(F 2 ) < ∞ 2 1,2 and E(DF H ) < ∞. Then F ∈ D .

232

Proof:

4. Transformations of the Wiener measure

For any h ∈ H and any smooth random variable G ∈ Sb we have E (DF, hH G) 1 = lim E ((F (ω + i(h)) − F (ω)) G) →0 2 2 1 = lim E F G(ω − i(h))eW (h)− 2 h H − F G →0 = E (F (GW (h) − DG, hH )) .

The result then follows from Lemma 1.3.1.

For any h ∈ H the random variable W (h) is a.s. H-continuously diﬀerentiable and D(W (h)) = h (see Exercise 4.1.3).

4.1.4 Transformations induced by elementary processes Denote by {ei , i ≥ 1} a complete orthonormal system in H. We will assume that ei ∈ Ω∗ ; in that way W (ei ) is a continuous linear functional on Ω. Consider a smooth elementary process u ∈ SH (see Section 1.3.1) of the form N ψ j (W (e1 ), . . . , W (eN ))ej , (4.7) u= j=1

where ψ ∈

Cb∞ (RN ; RN ).

The transformation T : Ω → Ω induced by u is

T (ω) = ω +

N

ψ j (W (e1 ), . . . , W (eN ))i(ej ).

j=1

Deﬁne the random variable ⎛ η(u) = | det(I + ∆)| exp⎝−

N

u, ej H W (ej ) −

j=1

where

N 1

2

⎞ u, ej 2H ⎠ ,

(4.8)

j=1

∆ = ψ (W (N ) )

and W (N ) = (W (e1 ), . . . , W (eN )). We claim that η can also be written as 1 η(u) = |det2 (I + Du)| exp(−δ(u) − u2H ), 2

(4.9)

where det2 (I + Du) denotes the Carleman-Fredholm determinant of the square integrable kernel I + Du (see the appendix, section A4). In order

4.1 Anticipating Girsanov theorems

233

to deduce (4.9), let us ﬁrst remark that for an elementary process u of the form (4.7) the Skorohod integral δ(u) can be evaluated as follows: δ(u) =

N

ψ j (W (N ) )W (ej ) −

j=1

=

N

N N ∂ψ j j=1 i=1

∂xi

(W (N ) )ei , ej H

u, ej H W (ej ) − T ∆.

(4.10)

j=1

On the other hand, the derivative Du is equal to Du =

N N

∆ij ej ⊗ ei ,

∆ij =

j=1 i=1

∂ψ j (W (N ) ), ∂xi

(4.11)

and the Carleman-Fredholm determinant of the kernel I + Du is det(I + ∆) exp(−T ∆), which completes the proof of (4.9). The following proposition contains the basic results on the problems (A) and (B) for the transformation T induced by u, when u ∈ SH . Proposition 4.1.4 Suppose that u is a smooth elementary process of the form (4.7). It holds that (i) If T is one to one, we have E[η(u)F (T )] ≤ E[F ]

(4.12)

for any positive and bounded random variable F . (ii) If det2 (I +Du) = 0 a.s. (which is equivalent to saying that the matrix I + ψ (x) is invertible a.e.), then P ◦ T −1 ≤ P.

(4.13)

E[η(u)F (T )] = E[F ],

(4.14)

(iii) If T is bijective, then

for any positive and bounded random variable F . Proof: Let us ﬁrst remark that T is one to one (onto) if and only if the mapping ϕ(x) = x + ψ(x) is one to one (onto) on RN . We start by showing (i). It suﬃces to assume that the random variable F has the form F = f (W (e1 ), . . . , W (eN ), G)

(4.15)

234

4. Transformations of the Wiener measure

where G denotes the vector (W (eN +1 ), . . . , W (eM )), M > N , and f belongs to the space Cb∞ (RM ). The composition F (T ) is given by F (T ) = f (W (e1 ) + u, e1 H , . . . , W (eN ) + u, eN H , G). Note that u, ej H = ψ j (W (N ) ). Applying the change-of-variables formula and using the one to one character of ϕ, we obtain ⎛ ⎞ N N 1 E[ηF (T )] = E[| det(I + ∆)| exp ⎝− u, ej H W (ej ) − u, ej 2H ⎠ 2 j=1 j=1 ×f (W (e1 ) + u, e1 H , . . . , W (eN ) + u, eN H , G)] ⎛ ⎞ N det I + ψ (x) exp⎝− 1 (xj + ψ j (x))2 ⎠ =E 2 j=1 RN ×f (x1 + ψ 1 (x), . . . , xN + ψ N (x), G)(2π)− 2 dx1 · · · dxN ⎞ ⎛ N N 1 =E exp⎝− yj2 ⎠ f (y1 , . . . , yN , G) (2π)− 2 dy1 · · · dyN 2 N ϕ(R ) j=1 ⎞ ⎛ N N 1 ≤E exp⎝− yj2 ⎠ f (y1 , . . . , yN , G)(2π)− 2 dy1 · · · dyN 2 N R j=1 N

= E(F ).

(4.16)

This concludes the proof of (i). Property (ii) follows from the fact that ϕ is locally one to one. Indeed, we can ﬁnd a countable family of open sets Gn ⊂ RN such that their union is {det(I + ψ ) = 0} and ϕ is one to one on each Gn . This implies by (i) that E[η(u)1{W (N )∈ Gn } F (T )] ≤ E[F ] for any nonnegative and bounded random variable F , which yields the desired absolute continuity. Finally, property (iii) is proved in the same way as (i). Remarks: Condition det2 (I + Du) = 0 a.s. implies a positive answer to problem (A). On the other hand, the bijectivity of T allows us to deduce a Girsanov-type theorem. Notice that (4.12) implies (4.13) and (4.12), and the additional hypothesis E[η(u)] = 1 implies (4.14). In the next section we will extend these results to processes that are not necessarily elementary.

4.1.5 Anticipating Girsanov theorems Let us ﬁrst consider the case of a process whose derivative is strictly less than one in the norm of H ⊗ H. We need the following approximation result.

4.1 Anticipating Girsanov theorems

235

Lemma 4.1.3 Let u ∈ L1,2 be such that DuH⊗H < 1 a.s. Then there exists a sequence of smooth elementary processes un ∈ SH such that a.s. un → u, η(un ) → η(u), and Dun H⊗H < 1 for all n. Proof: It suﬃces to ﬁnd a sequence un such that un converges to u in the norm of L1,2 , and Dun H⊗H ≤ 1. In fact, the almost sure convergence of un and η(un ) is then deduced by choosing a suitable subsequence (note that Du and δ(u) are continuous functions of u with respect to the norm of L1,2 , and η(u) is a continuous function of u, Du, and δ(u)). Moreover, to get the strict inequality DuH⊗H < 1, we replace un by (1 − n1 )un . The desired sequence un is obtained as follows. Fix a complete orthonormal system {ei , i ≥ 1} ⊂ Ω∗ in H, and denote by Fn the σ-ﬁeld generated by W (n) = (W (e1 ), . . . , W (en )). Deﬁne Pn u =

n

E[u, ei H |F Fn ]ei .

i=1

Then Pn u converges to u in the norm of L1,2 , and D[P Pn u]H⊗H ≤ E[Du|F Fn ]H⊗H ≤ 1. By Exercise 1.2.8 we know that E[u, ei H |F Fn ] = fi (W (n) ), where the func1,2 tion fi belongs to the Sobolev space W (Rn , N (0, In )). Replacing fi by ΨN ◦ fi , where ΨN ∈ C0∞ (R) is a function equal to x if |x| ≤ N and |ΨN (x)| ≤ 1, we can assume that fi and its partial derivatives are bounded. Finally, it suﬃces to smooth fi by means of the convolution with an approximation of the identity. In fact, we have n i,j=1

(ϕ ∗ ∂j fi )(x) ≤ 2

n

[ϕ ∗ (∂ ∂j fi ) ](x) ≤ 2

i,j=1

a.e., which allows us to complete the proof.

n

(∂ ∂j fi )2 (x) ≤ 1

i,j=1

The following result is due to Buckdahn and Enchev (cf. [48] and [91]). See also Theorem 6.1 in Kusuoka’s paper [178]. Proposition 4.1.5 Suppose that u ∈ L1,2 and DuH⊗H < 1 a.s. Then the transformation T (ω) = ω + i(u) veriﬁes (4.12), and P ◦ T −1 << P . Proof: Let un be the sequence provided by Lemma 4.1.3. The transformation T associated with each process un is one to one, because ϕn is a contraction (we assume Dun H⊗H ≤ 1 − n1 ). Then, by (4.12), for each n we have E[η(un )F (T Tn )] ≤ E[F ], for any nonnegative, continuous, and bounded function F . By Fatou’s lemma this inequality holds for the limit process u, and (4.12) is true. Finally,

236

4. Transformations of the Wiener measure

the absolute continuity follows from the fact that η(u) > 0 a.s. because DuH⊗H < 1 a.s. If in Proposition 4.1.5 we assume in addition that E[η(u)] = 1, then (4.14) holds. A suﬃcient condition for E[η(u)] = 1 has been given by Buckdahn in [48] (see also Exercise 4.1.4). In order to remove the hypothesis that the derivative is strictly bounded by one, we will show that if u is H-continuously diﬀerentiable then the corresponding transformation can be locally approximated by the composition of a linear transformation and a transformation induced by a process whose derivative is less than one. The H diﬀerentiability property seems to be fundamental in order to obtain this decomposition. The following two lemmas will be useful in completing the localization procedure. Lemma 4.1.4 Suppose that there exists a sequence of measurable sets Bn 1,2 and an element u ∈ Lloc such that ∪n Bn = Ω a.s., and E[η(u)1Bn F (T )] ≤ E[F ]

(4.17)

for any n ≥ 1, and for any nonnegative, measurable and bounded function F . Then (i) if det2 (I + Du) = 0 a.s., we have P ◦ T −1 << P ; (ii) if there exists a left inverse Tl such that Tl ◦ T = Id a.s., then E[η(u)F (T )] ≤ E[F ] for any nonnegative, measurable, and bounded function F . Proof: Part (i) is obvious. In order to show (ii) we can assume that the Tl−1 (Bn )), and we sets Bn are pairwise disjoint. We can write Bn = T −1 (T have E[η(u)1Bn F (T )] ≤ E[1T −1 (Bn ) F (T )] ≤ E[F ]. E[η(u)F (T )] = n

n

l

the class of a.s. H-continuously diﬀerentiable We will denote by processes (or H-valued random variables). 1 (H) CH

1 (H), u2 , u3 ∈ L1,2 , and denote by T1 , T2 , and Lemma 4.1.5 Let u1 ∈ CH T3 the corresponding transformations. Suppose that

(i) P ◦ T2−1 << P ; (ii) T3 = T1 ◦ T2 Then we have

(or u3 = u2 + u1 (T T2 )).

4.1 Anticipating Girsanov theorems

237

(a) I + Du3 = [I + (Du1 )(T T2 )] (I + Du2 ); (b) η(u3 ) = η(u1 )(T T2 )η(u2 ). Remarks:

We recall that the composition (Du1 )(Du2 ) veriﬁes

(Du1 )(Du2 ) =

∞

Dek (u1 , ei H )Dej (u2 , ek H )ei ⊗ ej .

i,j,k=1

Proof of Lemma 4.1.5:

From Lemma 4.1.2 we deduce

Du3 = Du2 + (Du1 )(T T2 ) + (Du1 )(T T2 )(Du2 ), which implies (a). In order to prove (b) we use the following property of the Carleman-Fredholm determinant: det2 [(I + K1 )(I + K2 )] = det2 (I + K1 )det2 (I + K2 ) exp(−T(K1 K2 )). Hence, η(u3 )

= det2 (I + (Du1 )(T T2 )) det2 (I + Du2 ) × exp(−T[(Du1 )(T T2 )(Du2 )]) & 1 × exp −δ(u2 ) − δ(u1 (T T2 )) − u2 2H 2 ' 1 − u1 (T T2 )2H − u2 , u1 (T T2 )H . 2

Finally, we use the property T2 )) δ(u1 (T

= δ(f (W (h) + h, u2 H )g) = f (W (h) + h, u2 H )W (g) − f (W (h) + h, u2 H )(h, gH −Du2 , h ⊗ gH ) = δ(u1 )(T T2 ) − u1 (T T2 ), u2 H − T[(Du1 )(T T2 )(Du2 )],

which allows us to complete the proof. We are now able to show the following result.

1 Proposition 4.1.6 Let u ∈ CH (H), and assume that det2 (I + Du) = 0 (which is equivalent to saying that I + Du is invertible a.s.). Then there exists a sequence of measurable subsets Gn ⊂ Ω such that ∪n Gn = Ω a.s., and three sequences of processes un,1 , un,2 , and un,3 such that

(a) un,1 = vn is deterministic; (b) un,2 belongs to L1,2 and Dun,2 H⊗H ≤ c < 1;

238

4. Transformations of the Wiener measure

(c) un,3 is a smooth elementary process given by a linear map ψ n such that det(I + ψ n ) = 0; (d) T = Tn,1 ◦ Tn,2 ◦ Tn,3 on Gn , where T and Tn,i (i ( = 1, 2, 3) are the transformations associated with the processes u and un,i , respectively. Tn,3 ) + un,3 , on Gn . In other words, u = vn + un,2 (T Proof: Let {ei , i ≥ 1} ⊂ Ω∗ be a complete orthonormal system in H. Fix a countable and dense subset H0 of H. Consider the set K ⊂ H ⊗ H formed by the kernels of the form n

K=

λij ei ⊗ ej ,

i,j=1

where n ≥ 1, λij ∈ Q, and det(I + λ) = 0. The set K is dense in the set {K ∈ H ⊗ H : det2 (I + K) = 0}, and for any K ∈ K the operator I + K has ˜ the smooth elementary process a bounded inverse. We will denote by K associated with K, namely, ˜ = K

n

λij W (ej )ei .

i,j=1

˜ = K(ei ), i ≥ 1. Fix a positive number 0 < a < c , where So, we have Dei K 9 0 < c < 1 is a ﬁxed number. Set γ(K) = (I + K)−1 −1 L(H,H) . Consider the set of triples ν = (K, v, n), with n ≥ 1, K ∈ K, and v ∈ H0 , n . This countable set will be denoted by I. For each such that γ(K) < 3a ν ∈ I we deﬁne the following set: 4 Cν = ω ∈ Ω : sup Du(ω + i(h)) − KH⊗H ≤ aγ(K), 1

h H ≤ n

aγ(K) 5 ˜ . u(ω) − K(ω) − vH ≤ n Then the union of the sets Cν when ν ∈ I is the whole space Ω a.s., because det2 (I + Du) = 0 and u is a.s. H-continuously diﬀerentiable. We can ﬁnd σ-compact sets Gν ⊂ Cν such that P (Gν ) = P (Cν ). The sets Gν constitute the family we are looking for, and we are going to check that these sets satisfy properties (a) through (d) for a suitable sequence of processes. Let φ ∈ C0∞ (R) be a nonnegative function such that |φ(t)| ≤ 1 and |φ (t)| ≤ 4 for all t, φ(t) = 1 for |t| ≤ 13 , and φ(t) = 0 for |t| ≥ 23 . Deﬁne uν,1 uν,3 uν,2

= v, ˜ = K, ! "! " −1 −1 −1 = φ nρGν (T Tν, Tν, Tν, 3 ) u(T 3 ) − uν,3 (T 3) − v .

4.1 Anticipating Girsanov theorems

239

Clearly, these uν,1 and uν,3 satisfy conditions (a) and (c), respectively. Then it suﬃces to check properties (b) and (d). The process uν,2 is the product of two factors: The ﬁrst one is a random variable that is bounded and belongs to D1,p for all p ≥ 2, due to property (iv) of the function ρ. The second factor is H-continuously diﬀerentiable. Let us now show that the derivative of uν,2 is bounded by c < 1 in the norm of H ⊗ H. Deﬁne ˜ − v]. Tν,3 ) = φ(nρGν )[u − K vν = uν,2 (T

(4.18)

−1 Then uν,2 = vν (T Tν, 3 ), and we have

Duν,2 H⊗H

−1 = [I + K]−1 (Dvν )(T Tν, 3 ) H⊗H −1 ≤ γ(K)−1 (Dvν )(T Tν, 3 )H⊗H .

So it suﬃces to check that Dvν H⊗H ≤ cγ(K). We have Dvν H⊗H

≤ 1{ρG

ν

1 (Du
− KH⊗H

+4nu − uν,3 − vH ). Suppose that ρGν (ω) < n1 . Then there exists an element h ∈ H such that ω + i(h) ∈ Gν and hH < n1 . Consequently, from the deﬁnition of the set Cν we obtain for this element ω (Du)(ω) − KH⊗H ≤ aγ(K), and u(ω) − uν,3 (ω) − vH

≤ u(ω + i(h)) − uν,3 (ω + i(h)) − vH +u(ω + i(h)) − u(ω) − K, hH 1 aγ(K) + ≤ (Du)(ω + ti(h)) − K, hH dtH n 0 2aγ(K) . ≤ n

Hence, we have Dvν H⊗H ≤ 9aγ(K). Also this argument implies that uν,2 is bounded. As a consequence (see Exercise 4.1.5), uν,2 belongs to L1,2 . Finally, we will check property (d). Notice ﬁrst that ω ∈ Gν implies ρGν (ω) = 0. Therefore, from (4.18) we obtain u(ω) = v + uν,2 (T Tν,3 (ω)) + uν,3 (ω). Using the above proposition we can prove the following absolute continuity result.

240

4. Transformations of the Wiener measure

Theorem 4.1.1 Let u be a process that is a.s. H-continuously diﬀerentiable. Suppose that I + Du is invertible a.s. (or det2 (I + Du) = 0 a.s.). Denote by T the transformation deﬁned by T (ω) = ω + i(u(ω)). Then P ◦ T −1 << P . Tn,2 ◦T Tn,3 and the sequence Proof: Consider the decomposition T = Tn,1 ◦T of sets Gn introduced in Proposition 4.1.6. The transformations Tn,i , i = 1, 2, 3, verify (4.12). For i = 1, 3 this follows from part (i) of Proposition 4.1.4, and for i = 2 it follows from Proposition 4.1.5. From Lemma 4.1.5 and using the local property of η(u), we have Tn,3 )η(un,3 ) on η(u) = η(v)(T Tn,2 ◦ Tn,3 )η(un,2 )(T

Gn .

Consequently, for any nonnegative and bounded random variable F we obtain E [1Gn F (T )η(u)] = E [1Gn F (T Tn,1 ◦ Tn,2 ◦ Tn,3 )η(v)(T Tn,2 ◦ Tn,3 )η(un,2 )(T Tn,3 )η(un,3 )] ≤ E [F (T Tn,1 ◦ Tn,2 )η(v)(T Tn,2 )η(un,2 )] ≤ E [F (T Tn,1 )η(v)] ≤ E[F ], and the result follows from Lemma 4.1.4. The main result of this section is the following version of Girsanov’s theorem due to Kusuoka ([178], Theorem 6.4). Theorem 4.1.2 Let u be a process that is H-continuously diﬀerentiable, and denote by T the transformation deﬁned by T (ω) = ω +i(u(ω)). Suppose that (i) T is bijective; (ii) I + Du is invertible a.s. (or det2 (I + Du) = 0 a.s.). Then there exists a probability Q equivalent to P , such that Q ◦ T −1 = P , given by 1 dQ = |det2 (I + Du)| exp(−δ(u) − u2H ). (4.19) dP 2 Proof: From Theorem 4.1.1 we know that Eq. (4.17) of Lemma 4.1.4 holds with the sequence of sets Gn given by Proposition 4.1.6. Then part (ii) of Lemma 4.1.4 yields E[η(u)F (T )] ≤ E[F ],

(4.20)

for any nonnegative and bounded random variable F . It is not diﬃcult to see, using the implicit function theorem, that the inverse transformation T −1 is also associated with an a.s. H-continuously diﬀerentiable process.

4.2 Markov random ﬁelds

241

Let us denote this process by v. From Lemma 4.1.5 applied to the composition I = T −1 ◦ T , we get 1 = η(v)(T )η(u). Therefore, P ◦ T << P , and applying (4.20) to T −1 we get E(F ) = E[η(v)η(u)(T −1 )F ] ≤ E[η(u)F (T )],

so the equality in (4.20) holds and the proof is complete.

Exercises 4.1.1 Show that the law of the continuous process Wt +tf (W W1 ) is absolutely continuous with respect to the Wiener measure on C0 ([0, 1]) if and only if Y + f (Y ) has a density when Y has N (0, 1) law. If f is locally Lipschitz, a suﬃcient condition for this is f (x) = −1 a.e. 4.1.2 Using Itˆo’s formula, show that S S 2W Wt Wt2 dW Wt − 2 dt ≤ e−1 , E exp − 2 4 0 (1 − t) 0 (1 − t) where S = inf {t : Wt2 = 1 − t}. 4.1.3 Show that for any h ∈ H the random variable W (h) is a.s. H continuously diﬀerentiable and D(W (h)) = h. 4.1.4 Let u ∈ L1,2 be a process satisfying the following properties: (i) Du H⊗H < c < 1; 1 (ii) E exp 2q 0 u2t dt < ∞ for some q > 1. Show that E(η) = 1 (see Enchev [91]). 1 4.1.5 Let F ∈ CH , and G ∈ D1,p for all p > 1. Assume that E(F 2 G2 ) < ∞ and E(DF 2H G2 + DG2H F 2 ) < ∞.

Show that F G belongs to D1,2 .

4.2 Markov random ﬁelds We start by introducing the notion of conditional independence. Deﬁnition 4.2.1 Let F1 , F2 , and F3 be three sub-σ-algebras in a probability space (Ω, F, P ). We will say that F1 and F2 are conditionally independent given F3 if P (A1 ∩ A2 |F F3 ) = P (A1 |F F3 )P (A2 |F F3 ) for all A1 ∈ F1 and A2 ∈ F2 . In this case we will write F1

F3

F2 .

242

4. Transformations of the Wiener measure

The conditional independence F1

F3

F2 is equivalent to the property

P (A1 |F F3 ) = P (A1 |F F2 ∨ F3 ) for all A1 ∈ F1 . We refer to Rozanov [296] for a detailed analysis of this notion and its main properties. The conditional independence allows us to introduce several deﬁnitions of Markov’s property. Let {X(t), t ∈ [0, 1]} be a continuous stochastic process. (a) We will say that X is a Markov process if for all t ∈ [0, 1] we have σ{Xr , r ∈ [0, t]}

σ{Xt }

σ{Xr , r ∈ [t, 1]}.

(b) We will say that X is a Markov random ﬁeld if for all 0 ≤ s < t ≤ 1 we have σ{Xr , r ∈ [s, t]}

σ{Xs ,Xt }

σ{Xr , r ∈ [0, 1] − (s, t)}.

Property (a) is stronger than (b) (see Exercise 4.2.3). The converse is not true (see Exercise 4.2.4). In the next section we will apply Theorem 4.1.2 to study the Markov ﬁeld property of solutions to stochastic diﬀerential equations with boundary conditions.

4.2.1 Markov ﬁeld property for stochastic diﬀerential equations with boundary conditions Let {W Wt , t ∈ [0, 1]} be a standard Brownian motion deﬁned on the canonical probability space (Ω, F, P ). Consider the stochastic diﬀerential equation 2

Xt = X0 −

t 0

f (Xs )ds + Wt

X0 = g(X1 − X0 ),

(4.21)

where f, g : R → R are two continuous functions. Observe that the periodic condition X0 = X1 is not included in this formulation. In order to handle this and other interesting cases, one should consider more general boundary conditions of the form X0 = g(e−λ X1 − X0 ), with λ ∈ R. The periodic case would correspond to λ = 0 and g(x) = (e−λ − 1)−1 x. In order to simplify the exposition we will assume henceforth that λ = 0.

4.2 Markov random ﬁelds

243

When f ≡ 0, the solution of (4.21) is Yt = Wt + g(W W1 ).

(4.22)

Denote by Σ the set of continuous functions x : [0, 1] → R such that x0 = g(x1 − x0 ). The mapping ω → Y (ω) is a bijection from Ω into Σ. Consider the process Y = {Y Yt , t ∈ [0, 1]} given by (4.22). Deﬁne the transformation T : Ω → Ω by t

T (ω)t = ω t +

f (Y Ys (ω))ds.

(4.23)

0

Lemma 4.2.1 The transformation T is a bijection of Ω if and only if Eq. (4.21) has a unique solution for each ω ∈ Ω; in this case this solution is given by X = Y (T −1 (ω)). Proof: If T (η) = ω, then the function Xt = Yt (η) solves Eq. (4.21) for Wt = ω t . Indeed: t t f (Y Ys (η))ds = X0 + Wt − f (Xs )ds. Xt = X0 + η t = X 0 + ω t − 0

0

Conversely, given a solution X to Eq. (4.21), we have T (Y −1 (X)) = W . Indeed, if we set Y −1 (X) = η, then t t T (η)t = η t + f (Y Ys (η))ds = η t + f (Xs )ds 0

= η t + Wt − Xt + X0 = Wt .

0

There are suﬃcient conditions for T to be a bijection (see Exercise 4.2.10). Henceforth we will impose the following assumptions: (H.1) There exists a unique solution to Eq. (4.21) for each ω ∈ Ω. (H.2) f and g are of class C 1 . Now we turn to the discussion of the Markov ﬁeld property. First notice that the process Y is a Markov random ﬁeld (see Exercise 4.2.3). Suppose that Q is a probability on Ω such that P = Q◦T −1 . Then {T (ω)t , 0 ≤ t ≤ 1} will be a Wiener process under Q, and, consequently, the law of the process X under the probability P coincides with the law of Y under Q. In this way we will translate the problem of the Markov property of X into the problem of the Markov property of the process Y under a new probability Q. This problem can be handled, provided Q is absolutely continuous with respect to the Wiener measure P and we can compute an explicit expression for its Radon-Nikodym derivative. To do this we will make use of Theorem 4.1.2, applied to the process Yt ). (4.24) ut = f (Y

244

4. Transformations of the Wiener measure

Notice that T is bijective by assumption (H.1) and that u is H-continuously diﬀerentiable by (H.2). Moreover, Ds ut = f (Y Yt )[g (W W1 ) + 1{s≤t} ].

(4.25)

The the Carleman-Fredholm determinant of kernel (4.25) is computed in the next lemma. Yt ). Then Lemma 4.2.2 Set αt = f (Y 1 1 det2 (I + Du) = 1 + g (W W1 ) 1 − e− 0 αt dt e−g (W1 ) 0 αt dt . Proof:

From (A.12) applied to the kernel Du, we obtain ∞ γn , det2 (I + Du) = 1 + n! n=2

where

(4.26)

γn

= [0,1]n

det(1{i= j} Dti utj )dt1 . . . dtn

= n!

{t1
=

α(t)dt

det(1{i= j} Dti utj )dt1 . . . dtn

det Bn ,

0

and the matrix Bn is given by ⎡ 0 g (W W1 ) + 1 g (W W1 ) + 1 ⎢ g (W W ) 0 g (W W1 ) + 1 1 ⎢ (W W ) g (W W ) 0 g B=⎢ 1 1 ⎢ ⎣ ··· ··· ··· W1 ) g (W W1 ) g (W W1 ) g (W

··· ··· ··· ··· ···

g (W W1 ) + 1 g (W W1 ) + 1 g (W W1 ) + 1 ··· 0

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

Simple computations show that for all n ≥ 1 det Bn = (−1)n g (W W1 )n (g (W W1 ) + 1) + (−1)n+1 g (W W1 )(g (W W1 ) + 1)n . Hence, det2 (I + Du) = 1 +

n 1 ∞ 1 α(t)dt n! 0 n=1

" ! × (−1)n g (W W1 )n (g (W W1 ) + 1) + (−1)n+1 g (W W1 )(g (W W1 ) + 1)n

1

W1 ) + 1)e−g (W1 ) 0 αt dt − g (W W1 )e−(g (W1 )+1) = (g (W 1 1 = 1 + g (W W1 ) 1 − e− 0 αt dt e−g (W1 ) 0 αt dt .

1 0

αt dt

4.2 Markov random ﬁelds

245

Therefore, the following condition implies that det2 (I + Du) = 0 a.s.: (H.3) 1 + g (y) 1 − e−f (x+g(y)) = 0, for almost all x, y in R. Suppose that the functions f and g satisfy conditions (H.1) through (H.3). Then the process u given by (4.24) satisﬁes the conditions of Theorem 4.1.2, and we obtain & 1 ' dQ (4.27) η(u) = = 1 + g (W W1 ) 1 − exp − f (Y Yt )dt dP 0 1 1 × exp −g (W W1 ) f (Y Yt )dt − f (Y Yt )dW Wt −

1 2

0

1

0

f (Y Yt )2 dt .

0

We will denote by Φ the term Φ = 1 + g (W W1 ) 1 − exp −

&

1

f (Y Yt )dt

' ,

0

and let L be the exponential factor in (4.27). Using the relationship between the Skorohod and Stratonovich integrals, we can write 1 1 1 1 1 f (Y Yt )dW Wt = f (Y Yt ) ◦ dW Wt − f (Y Yt )dt − g (W W1 ) f (Y Yt )dt. 2 0 0 0 0 Consequently, the term L can be written as 1 1 1 1 1 L = exp − f (Y Yt ) ◦ dW Wt + f (Y Yt )dt − f (Y Yt )2 dt . 2 0 2 0 0

(4.28)

In this form we get η(u) = |Φ|L. The main result about the Markov ﬁeld property of the process X is the following: Theorem 4.2.1 Suppose that the functions f and g are of class C 2 and f has linear growth. Suppose furthermore that the equation t Xt = X0 − 0 f (Xs )ds + Wt (4.29) X0 = g(X1 − X0 ) has a unique solution for each W ∈ Ω and that (H.3) holds. Then the process X veriﬁes the Markov ﬁeld property if and only if one of the following conditions holds: (a) f (x) = ax + b, for some constants a, b ∈ R,

246

4. Transformations of the Wiener measure

(b) g = 0, (c) g = −1. Remarks: 1. If condition (b) or (c) is satisﬁed, we have an initial or ﬁnal ﬁxed value. In this case, assuming only that f is Lipschitz, it is well known that there is a unique solution that is a Markov proces (not only a Markov random ﬁeld). 2. Suppose that (a) holds, and assume that the implicit equation x = g((e−a − 1)x + y) has a unique continuous solution x = ϕ(y). Then Eq. (4.29) admits a unique solution that is a Markov random ﬁeld (see Exercise 4.2.6). Proof: Taking into account the above remarks, it suﬃces to show that if X is a Markov random ﬁeld then one of the above conditions is satisﬁed. Let Q be the probability measure on C0 ([0, 1]) given by (4.27). The law of the process X under P is the same as the law of Y under Q. Therefore, Y is a Markov ﬁeld under Q. For any t ∈ (0, 1), we deﬁne the σ-algebras 9 : 9 : W1 ) , Fti = σ Yu , 0 ≤ u ≤ t = σ Wu , 0 ≤ u ≤ t, g(W : 9 9 : Fte = σ Yu , t ≤ u ≤ 1, Y0 = σ Wu , t ≤ u ≤ 1 , and Ft0 = σ{Y Y0 , Yt } = σ{W Wt , g(W W1 )}. The random variable L deﬁned in (4.28) can be written as L = Lit Let , where t 1 t 1 t Lit = exp − f (Y Ys ) ◦ dW Ws + f (Y Ys )ds − f (Y Ys )2 ds 2 0 2 0 0 and Let = exp −

1

f (Y Ys ) ◦ dW Ws +

t

1 2

t

1

f (Y Ys )ds −

1 2

1

f (Y Ys ))2 ds .

t

Notice that Lit is Fti -measurable and Let is Fte -measurable. For any nonnegative random variable ξ, Fti -measurable, we deﬁne (see Exercise 4.2.11) ∧ξ = EQ (ξ|F Fte ) =

E(ξ|Φ|Lit | Fte ) E(ξη(u) | Fte ) = . e E(η(u) | Ft ) E(|Φ|Lit | Fte )

The denominator in the above expression is ﬁnite a.s. because η(u) is integrable with respect to P . The fact that Y is a Markov ﬁeld under Q implies that the σ-ﬁelds Fti and Fte are conditionally independent given Ft0 . As a consequence, ∧ξ is Ft0 -measurable. Choosing ξ = (Lit )−1 and ξ = χ(Lit )−1 ,

4.2 Markov random ﬁelds

247

where χ is a nonnegative, bounded, and Fti -measurable random variable, we obtain that E(|Φ| | Fte ) E(|Φ|Lit | Fte )

and

E(χ|Φ| | Fte ) E(|Φ|Lit | Fte )

are Ft0 -measurable. Consequently, Gχ =

E(χ|Φ| | Fte ) E(|Φ| | Fte )

is also Ft0 -measurable. The next step will be to translate this measurability property into an analytical condition using Lemma 1.3.3. First notice that if χ is a smooth random variable that is bounded and has a bounded derivative, then Gχ 1,2 belongs to Dloc because f has linear growth. Applying Lemma 1.3.3 to the Wt , W1 } yields random variable Gχ and to the σ-ﬁeld σ{W d Ds [Gχ ] = 0 ds d a.e. on [0, 1]. Notice that ds Ds χ = 0 a.e. on [t, 1], because χ is Fti -measurable (again by Lemma 1.3.3). Therefore, for almost all s ∈ [t, 1], we get & & ' ' d d e e e e Ds |Φ| | Ft . E χ Ds |Φ| | Ft E [|Φ| | Ft ] = E [χ|Φ| | Ft ] E ds ds

The above equality holds true if χ is Fte -measurable. So, by a monotone class argument, it holds for any bounded and nonnegative random variable χ, and we get that W1 )Zf (W Ws + g(W W1 )) −g (W 1 d Ds Φ = Φ ds 1 + g (W W1 )(1 − Z) is Fte -measurable for almost all s ∈ [t, 1] (actually for all s ∈ [t, 1] by continuity), where 1 f (W Wr + g(W W1 ))dr . Z = exp − 0

Suppose now that condition (a) does not hold, that is, there exists a point y ∈ R such that f (y) = 0. By continuity we have f (x) = 0 for all x in some interval (y − , y + ). Given t < s < 1, deﬁne Ws + g(W W1 )) ∈ (y − , y + )}. As = {f (W Then P (As ) > 0, and 1As

g (W W1 )Z 1 + g (W W1 )(1 − Z)

248

4. Transformations of the Wiener measure

is Fte -measurable. Again applying Lemma 1.3.3, we obtain that & ' g (W W1 )Z d Dr =0 dr 1 + g (W W1 )(1 − Z) for almost all r ∈ [0, t] and ω ∈ As . This implies g (W W1 )(1 + g (W W1 ))f (W Wr + g(W W1 )) = 0 a.e. on [0, t] × As . Now, if B = As ∩ {f (W Wr + g(W W1 )) ∈ (y − , y + )}, we have that P (B) = 0 and W1 )(1 + g (W W1 )) = 0, g (W a.s. on B. Then if (b) and (c) do not hold, we can ﬁnd an interval I such W1 )(1 + g (W W1 )) = 0. The set B ∩ {W W1 ∈ I} has that if W1 ∈ I then g (W nonzero probability, and this implies a contradiction. Consider the stochastic diﬀerential equation (4.21) in dimension d > 1. One can ask under which conditions the solution is a Markov random ﬁeld. This problem is more diﬃcult, and a complete solution is not available. First we want to remark that, unlike in the one-dimensional case, the solution can be a Markov process even though f is nonlinear. In fact, suppose that the boundary conditions are of the form X0ik

= ak ;

1 ≤ k ≤ l,

X1jk

= bk ;

1 ≤ k ≤ d − l,

where {i1 , . . . , il } ∪ {j1 , . . . , jd−l } is a partition of {1, . . . , d}. Assume in addition that f is triangular, that means, f k (x) is a function of x1 , . . . , xk for all k. In this case, if for each k, f k satisﬁes a Lipschitz and linear growth condition on the variable xk , one can show that there exists a unique Wt with the above boundary solution of the equation dXt + f (Xt ) = dW conditions, and the solution is a Markov process. The Markov ﬁeld property for triangular functions f and triangular boundary conditions has been studied by Ferrante [97]. Other results in the general case obtained by a change of probability argument are the following: (1) In dimension one, and assuming a linear boundary condition of the F1 X1 = h0 , Donati-Martin (cf. [80]) has obtained the existence type F0 X0 +F and uniqueness of a solution for the equation dXt = σ(Xt ) ◦ dW Wt + b(Xt )

4.2 Markov random ﬁelds

249

when the coeﬃcients b and σ are of class C 4 with bounded derivatives, and F0 F1 = 0. On the other hand, if σ is linear (σ(x) = αx), h0 = 0, and assuming that b is of class C 2 , then one can show that the solution X is a Markov random ﬁeld only if the drift is of the form b(x) = Ax + Bx log |x|, where |B| < 1. See also [5] for a discussion of this example using the approach developed in Section 4.2.3. (2) In the d-dimensional case one can show the following result, which is similar to Theorem 2.1 (cf. Ferrante and Nualart [98]): Theorem 4.2.2 Suppose f is inﬁnitely diﬀerentiable, g is of class C 2 , and W1 )+g (W W1 )) = 0 a.s., where φ(t) is the solution of the linear det(I−φ(1)g (W Yt )φ(t)dt, φ(0) = I. We also assume that the equation equation dφ(t) = f (Y

t Xt = X0 − 0 f (Xs )ds + Wt X0 = g(X1 − X0 )

has a unique solution for each W ∈ C0 ([0, 1]; Rd ), and that the following condition holds: (H.4) span ∂ ∂i1 · · · ∂im f (x); i1 , . . . , im ∈ {1, . . . , d}, m ≥ 1 = Rd×d , for all x ∈ Rd . Then we have that g (x) is zero or −IId , that is, the boundary condition is of the form X0 = a or X1 = b. (3) It is also possible to have a dichotomy similar to the one-dimensional case in higher dimensions (see Exercise 4.2.12).

4.2.2

Markov ﬁeld property for solutions to stochastic partial diﬀerential equations

In this section we will review some results on the germ Markov ﬁeld (GMF) property for solutions to stochastic partial diﬀerential equations driven by a white noise which have been obtained by means of the technique of change of probability. Let D be a bounded domain in Rk with smooth boundary, and consider a continuous stochastic process X = {Xz , z ∈ D}. We will say that X is a germ Markov ﬁeld (GMF) if for any > 0 and any open subset A ⊂ D, the σ-ﬁelds σ{Xz , z ∈ A} and σ{Xz , z ∈ D − Ac } are conditionally independent given the σ-ﬁeld σ{Xz , z ∈ (∂A) }, where (∂A) denotes the -neighborhood of the boundary of A. We will ﬁrst discuss in some detail the case of an elliptic stochastic partial diﬀerential equation with additive white noise.

250

4. Transformations of the Wiener measure

(A) Elliptic stochastic partial diﬀerential equations Let D be a bounded domain in Rk with smooth boundary, and assume k = 1, 2, 3. Let λk denote the Lebesgue measure on D, and set H = L2 (D, B(D), λk ). Consider an isonormal Gaussian process W = {W (h), h ∈ H} associated with H. That is, if we set W (A) = W (1A ), then W = {W (A), A ∈ B(D)} is a zero-mean Gaussian process with covariance E(W (A)W (B)) = λk (A ∩ B). We want to study the equation ˙ (x), −∆U (x) + f (U (x)) = W U |∂D = 0.

x ∈ D,

(4.30)

Let us ﬁrst introduce the notion of the solution to (4.30) in the sense of distributions. Deﬁnition 4.2.2 We will say that a continuous process U = {U (x), x ∈ D} that vanishes in the boundary of D is a solution to (4.30) if ϕ(x)W (dx) −U, ∆ϕH + f (U ), ϕH = D ∞

for all ϕ ∈ C (D) with compact support. We will denote by G(x, y) the Green function associated with the Laplace operator ∆ with Dirichlet boundary conditions on D. That is, for any ϕ ∈ L2 (D), the elliptic linear equation −∆ψ(x) = ϕ(x), x ∈ D, (4.31) ψ |∂D = 0 possesses a unique solution in the Sobolev space H01 (D), which can be written as G(x, y)ϕ(y)dy. ψ(x) = D

We recall that H01 (D) denotes the completion of C0∞ (D) for the Sobolev norm · 1,2 . We will use the notation ψ = Gϕ. We recall that G is a symmetric function such that G(x, ·) is harmonic on D − {x}. One can easily show (see [54]) that U is a solution to the elliptic equation (4.30) if and only if it satisﬁes the integral equation G(x, y)f (U (y))dy = G(x, y)W (dy). (4.32) U (x) + D

D

Note that the right-hand side of (4.32) is a well-deﬁned stochastic integral because the Green function is square integrable. More precisely, we have G2 (x, y)dy < ∞. (4.33) sup x∈D

D

4.2 Markov random ﬁelds

251

In dimension k > 3 this property is no longer true, and for this reason the analysis stops at dimension three. We will denote by U0 the solution of (4.30) for f = 0, that is, G(x, y)W (dy). (4.34) U0 (x) = D

Using Kolmogorov’s criterion one can show (see Exercise 4.2.13) that the ¨older continuous process {U U0 (x), x ∈ D} has Lipschitz paths if k = 1, H¨ paths of order 1 − if k = 2, and Holder ¨ continuous paths of order 38 − if k = 3, for any > 0. The following result was established by Buckdahn and Pardoux in [54]. Theorem 4.2.3 Let D be a bounded domain of Rk , k = 1, 2, 3, with a smooth boundary. Let f be a continuous and nondecreasing function. Then Eq. (4.32) possesses a unique continuous solution. A basic ingredient in the proof of this theorem is the following inequality: Lemma 4.2.3 There exists a constant a > 0 such that for any ϕ ∈ L2 (D), Gϕ, ϕH ≥ aGϕ2H .

(4.35)

Proof: Set ψ = Gϕ. Then ψ solves Eq. (4.31). Multiplying this equation by ψ and integrating by parts, we obtain

k

∂ψ 2

∂xi = ϕ, ψH . H i=1 From Poincare’s ´ inequality (cf. [120, p. 157]) there exists a constant a > 0 such that for any ψ ∈ H01 (D),

k

∂ψ 2 2

∂xi ≥ aψH . H i=1 The result follows. We are going to reformulate the above existence and uniqueness theorem in an alternative way. Consider the Banach space B = {ω ∈ C(D), ω |∂D = 0}, equipped with the supremum norm, and the transformation T : B → B given by G(x, y)f (ω(y))dy. (4.36) T (ω)(x) = ω(x) + D

Note that {U (x), x ∈ D} is a solution to (4.32) if and only if T (U (x)) = U0 (x). Then Theorem 4.2.3 is a consequence of the following result.

252

4. Transformations of the Wiener measure

Lemma 4.2.4 Let f be a continuous and nondecreasing function. Then the transformation T given by (4.36) is bijective. Proof: Let us ﬁrst show that T is one to one. Let u, v ∈ B such that T (u) = T (v). Then u − v + G[f (u) − f (v)] = 0.

(4.37)

Multiplying this equation by f (u) − f (v), we obtain u − v, f (u) − f (v)H + G[f (u) − f (v)], f (u) − f (v)H = 0. Using the fact that f is nondecreasing, and Lemma 4.2.3, it follows that aG[f (u) − f (v)]2H ≤ 0. By (4.37) this is equivalent to au − v2H ≤ 0, so u = v and T is one to one. In order to show that T is onto, we will assume that f is bounded. The general case would be obtained by a truncation argument. Let v ∈ B, and let {vn , n ∈ N} be a sequence of functions in C 2 (D), with compact support in D, such that v − vn ∞ tends to zero as n tends to inﬁnity. Set hn = −∆vn . It follows from Lions ([199], Theorem 2.1, p. 171) that the elliptic partial diﬀerential equation −∆un + f (un ) = hn un |∂D = 0 admits a unique solution un ∈ H01 (D). Then, un + G[f (un )] = Ghn = vn ,

(4.38)

that is, T (un ) = vn . We now prove that un is a Cauchy sequence in L2 (D). Multiplying the equation un − um + G[f (un ) − f (um )] = vn − vm by f (un ) − f (um ), and using Lemma 4.2.3 and the monotonicity property of f , we get aG[f (un ) − f (um )]2H ≤ vn − vm , f (un ) − f (um )H , which implies, using the above equation, aun − um H ≤ vn − vm , f (un ) − f (um ) + 2a(un − um )H . Since {vn } is a Cauchy sequence in L2 (D) and f is bounded, {un } is a Cauchy sequence in L2 (D). Deﬁne u = lim un . Then f (un ) converges to f (u) in L2 (D). Taking the limit in (4.38), we obtain u + G[f (u)] = v. Thus u ∈ B (f is bounded) and T (u) = v.

4.2 Markov random ﬁelds

253

Let us now discuss the germ Markov ﬁeld property of the process U (x). First we will show that the Gaussian process U0 (x) veriﬁes the germ Markov ﬁeld property. To do this we shall use a criterion expressed in terms of the reproducing kernel Hilbert space (RKHS) H associated to U0 . Let H1 ⊂ L2 (Ω) be the Gaussian space (i.e., the ﬁrst chaos) generated by W . An element v ∈ B belongs to the RKHS H iﬀ there exists a random variable X ∈ H1 such that v(x) = E[XU U0 (x)], for all x ∈ D, i.e., iﬀ there exists φ ∈ L2 (D) such that v = Gφ. In other words, H = {v ∈ B : ∆v ∈ L2 (D)}, and v1 , v2 H = ∆v1 , ∆v2 H . We now have the following result (see Pitt [285] and K¨ unsch [177]). Proposition 4.2.1 A continuous Gaussian ﬁeld U = {U (x), x ∈ D} possesses the germ Markov ﬁeld property iﬀ its RKHS H ⊂ B is local in the sense that it satisﬁes the following two properties: (i) Whenever u, v in H have disjoint supports, u, vH = 0. (ii) If v ∈ H is of the form v = v1 + v2 with v1 , v2 ∈ B with disjoint supports, then v1 , v2 ∈ H. The RKHS associated to the process U0 veriﬁes conditions (i) and (ii), and this implies the germ Markov ﬁeld property of U0 . Concerning the process U , one can prove the following result. Theorem 4.2.4 Assume that f is a C 2 function such that f > 0 and f has linear growth. Then the solution {U (x), x ∈ D} of the elliptic equation (4.30) has the germ Markov property if and only if f = 0. This theorem has been proved by Donati-Martin and Nualart in [82]. In dimension one, Eq. (4.30) is a second-order stochastic diﬀerential equation studied by Nualart and Pardoux in [251]. In that case the germ σ-ﬁeld corresponding to the boundary points {s, t} is generated by the variables {Xs , X˙ s , Xt , X˙ t }, and the theorem holds even if the function f depends on Xt and X˙ t (assuming in that case more regularity on f ). The main diﬀerence between one and several parameters is that in dimension one, one can explicitly compute the Carleman-Fredholm determinant of Du. Similar to the work done in [81] and [82] we will give a proof for the case k = 2 or k = 3. Proof of Theorem 4.2.4: The proof follows the same lines as the proof of Theorem 2.1. We will indicate the main steps of the argument. Step 1: We will work on the abstract Wiener space (B, H, µ), where µ is the law of U0 , and the continuous injection i : H → B is deﬁned as follows: i(h)(x) = G(x, y)h(y)dy. D

254

4. Transformations of the Wiener measure

From Lemma 4.2.3 we deduce that i is one to one, and from Eq. (4.33) we see that i is continuous. The image i(H) is densely included in B. We identify H and H ∗ , and in this way B ∗ can be viewed as a dense subset of H, the inclusion map being given by G(y, ·)α(dy) = G∗ α. α→ D ∗

Finally, for any α ∈ B we have & ' eiα,ω µ(dω) = E exp(i U0 (x)α(dx)) B D & ' = E exp(i G(x, y)dW Wy α(dx)) D D 2 1 G(x, y)α(dx) dy = exp − 2 D D 1 ∗ 2 = exp − G αH , 2 which implies that (B, H, µ) is an abstract Wiener space. Note that i(H) coincides with the RKHS H introduced before, and that U0 (x) = ω(x) is now the canonical process in the space (B, B(B), µ). We are interested in the germ Markov ﬁeld property of the process U0 )(x). Let ν be the probability on B deﬁned by µ = ν ◦ T −1 . U (x) = T −1 (U That is, ν is the law of U . Step 2: Let us show that the transformation T veriﬁes the hypotheses of Theorem 4.1.2. We already know from Lemma 4.2.4 that T is bijective. Notice that we can write T (ω) = ω + i(f (ω)), so we have to show that: (i) the mapping ω → i(f (ω)) from B to H is H-continuously diﬀerentiable; (ii) the mapping IH + Du(ω) : H → H is invertible for all ω ∈ B, where Du(ω) is the Hilbert-Schmidt operator given by the kernel Du(ω)(x, y) = f (ω(x))G(x, y). Property (i) is obvious and to prove (ii), from the Fredholm alternative, it suﬃces to check that −1 is not an eigenvalue of Du(ω). Let h ∈ H be an element such that G(x, y)h(y)dy = 0. h(x) + f (ω(x)) D

4.2 Markov random ﬁelds

Multiplying this equality by D

h(x) f (ω(x))

255

and integrating over D, we obtain

h2 (x) dx + h, GhH = 0. f (ω(x))

From Lemma 4.2.3, h, GhH ≥ aGh2H , thus GhH = 0 and h = 0. Therefore, by Theorem 4.1.2 we obtain dν = |det2 (I + Du)| exp(−δ(u) − 12 u2H ). dµ

(4.39)

Set L = exp(−δ(u) − 12 u2 ). Step 3: For a ﬁxed domain A with smooth boundary Γ and such that A ⊂ D, we denote F i = σ{U U0 (x), x ∈ A},

F e = σ{U U0 (x), x ∈ D − A},

and F 0 = ∩>0 σ{U U0 (x), x ∈ (∂A) }. Consider the factorization L = Li Le , where 1 i 2 L = exp −δ(u1A ) − u1A H 2 and

1 2 L = exp −δ(u1D−A ) − u1D−A H . 2 e

We claim that J i is F i -measurable and J e is F e -measurable. This follows from the fact that the Skorohod integrals δ(u1A ) = f (U U0 (x))W (dx) A

and δ(u1D−A ) =

D −A

f (U U0 (x))W (dx)

are F i -measurable and F e -measurable, respectively (see [81]). Step 4: From Step 3 it follows that if f = 0, the Radon-Nikodym density given by (4.39) can be expressed as the product of two factors, one being F i -measurable, and the second one being F e -measurable. This factorization implies the germ Markov ﬁeld property of X under µ. Step 5: Suppose conversely that U possesses the germ Markov property under µ. By the same arguments as in the proof of Theorem 2.1 we can

256

4. Transformations of the Wiener measure

show that for any nonnegative random variable ξ that is F i -measurable, the quotient E[ξΦ | F e ] Gξ = E[Φ | F e ] is F 0 -measurable, where Φ = det2 (I + f (U U0 (x))G(x, y)). Observe that Φ ≥ 0 because the eigenvalues of the kernel f (U U0 (x))G(x, y)) are positive. Step 6: The next step will be to translate the above measurability property into an analytical condition. Fix > 0 such that A− = A − Γ and = (D−A)−Γ are nonempty sets. We have that G is σ{U U0 (x), x ∈ Γ }A+ ξ measurable. If we assume that ξ is a smooth random variable, then Gξ is 1,2 in Dloc , and by Lemma 1.3.3 we obtain that DGξ ∈ G(x, ·), x ∈ Γ H . This implies that for any function φ ∈ C0∞ (D − Γ ) we have φ, ∆DGξ H = 0 a.s. Suppose that φ ∈ C0∞ (A+ ). In that case we have in addition that φ, ∆DξH = 0, because ξ is F i -measurable. Consequently, for such a function φ we get E [ξφ, ∆DΦH | F e ] E[Φ | F e ] = E [ξΦ | F e ] E[φ, ∆DφH | F e ]. The above equality holds true for any bounded and nonnegative random variable ξ, therefore, we obtain that 1 φ, ∆DΦH Φ

(4.40)

is F e -measurable. Step 7: The derivative of the random variable Φ can be computed using the expression for the derivative of the Carleman-Fredholm determinant. We have U0 (x))G(x, y))−1 − I)Dz [f (U U0 (x))G(x, y)] . Dz Φ = ΦT ((I + f (U So, from (4.40) we get that U0 (x))G(x, y))−1 − I)φ, ∆D· [f (U U0 (x))G(x, y)]H T ((I + f (U is F e -measurable. Note that U0 (x))G(x, y)] = f (U U0 (x))G(x, z)G(x, y) Dz [f (U and ∆φ, G(x, ·)H = φ. Thus, we have that U0 (x))G(x, y))−1 − I)(φ(x)f (U U0 (x))G(x, y)) T ((I + f (U

4.2 Markov random ﬁelds

257

is F e -measurable, and we conclude that for any x ∈ A+ , f (U U0 (x)) K(y, x)G(x, y)dy D

is F e -measurable, where K(x, y) = ((I + Du)−1 − I)(x, y). Suppose now that there exists a point b ∈ R such that f (b) = 0. Then f will be nonzero in some interval J. Set U0 (x)) ∈ J}. A = {ω ∈ B : f (U The set A has nonzero probability, it belongs to F e , and 1A K(y, x)G(x, y)dy D

is F e -measurable. Applying again Lemma 1.3.3 and using the same arguments as above, we obtain that on the set A U0 (x1 ))ψ(x1 )G(x1 , x2 )](I + Du)−1 (x, x) = 0 G(I + Du)−1 [f (U for any function ψ ∈ C0∞ (A− ). So we get 1{f (U0 (x))∈J} 1{f (U0 (x1 ))∈J} G(I + Du)−1 (x, x1 )G(I + Du)−1 (x1 , x) = 0 − for all x, x1 such that x ∈ A+ and x1 ∈ A . Notice that the operator G(I + Du)−1 has a singularity in the diagonal of the same type as G. So from the above equality we get

1{f (U0 (x))∈J} = 0,

which is not possible. (B) Parabolic stochastic partial diﬀerential equations Consider the following equation studied in Section 2.4.2: 2 ∂u ∂2u ∂2W (t, x) ∈ [0, T ] × [0, 1], ∂t − ∂x2 = f (u(t, x)) + ∂t∂x , u(t, 0) = u(t, 1) = 0,

0 ≤ t ≤ T.

We will impose two diﬀerent types of boundary conditions: (B.1)

u(0, x) = u0 (x),

(B.2)

u(0, x) = u(1, x),

0 ≤ x ≤ 1.

In case (B.1) we are given a initial condition u0 ∈ C([0, 1]) such that u0 (0) = u0 (1) = 0, and in case (B.2) we impose a periodic boundary condition in time. Under some hypotheses on the function f there exists a unique continuous solution of the corresponding integral equation:

258

4. Transformations of the Wiener measure

For (B.1) a suﬃcient condition is that f is Lipschitz (see Theorem 2.4.3). For (B.2) (see [253]) we require that there exists a constant 0 < c < 2 such that (z − y)(f (z) − f (y)) ≤ c(z − y)2 for all y, z ∈ R. From the point of view of the Markov property of the solution, the behavior of these equations is completely diﬀerent. In case (B.1) the GMF property always holds. On the other hand, assuming that f (z) is of class Cb2 and that the boundary condition (B.2) holds, then the solution u has the GMF property if and only if f = 0. These results have been proved under more general assumptions on the function f in [253].

4.2.3 Conditional independence and factorization properties In this section we prove a general characterization of the conditional independence and apply it to give an alternative proof of Theorem 4.2.1. More precisely, we discuss the following general problem: Consider two independent sub-σ-ﬁelds F1 , F2 of a probability space, and let X and Y be two random variables determined by a system of the form X = g1 (Y, ω) Y = g2 (X, ω) where gi (y, ·) is Fi -measurable (i = 1, 2). Under what conditions on g1 and g2 are F1 and F2 conditionally independent given X and Y ? We will see that this problem arises in a natural way when treating stochastic equations with boundary conditions. Let (Ω, F, P ) be a complete probability space and let F1 and F2 be two independent sub-σ-ﬁelds of F. Consider two functions g1 , g2 : R × Ω → R such that gi is B(R) ⊗ Fi -measurable, for i = 1, 2, and that they verify the following conditions for some ε0 > 0: H1 For every x ∈ R and y ∈ R the random variables g1 (y, ·) and g2 (x, ·) possess absolutely continuous laws and the function δ(x, y) = sup 0<ε<ε0

1 P {|x − g1 (y)| < ε, |y − g2 (x)| < ε} ε2

is locally integrable in R2 . H2 For almost all ω ∈ Ω and for any |ξ| < ε0 , |η| < ε0 the system x − g1 (y, ω) = ξ (4.41) y − g2 (x, ω) = η has a unique solution (x, y) ∈ R2 .

4.2 Markov random ﬁelds

259

H3 For almost all ω ∈ Ω the functions y → g1 (y) and x → g2 (x) are continuously diﬀerentiable and there exists a nonnegative random variable H such that E(H) < ∞ and sup |y−g2 (x)|<ε0 |x−g1 (y)|<ε0

|1 − g1 (y)g2 (x)|

−1

≤ H

a.s.

Hypothesis H2 implies the existence of two random variables X and Y determined by the system X(ω) = g1 (Y (ω), ω) (4.42) Y (ω) = g2 (X(ω), ω). Theorem 4.2.5 Let g1 and g2 be two functions satisfying hypotheses H1 through H3. Then the following statements are equivalent: (i) F1 and F2 are conditionally independent given the random variables X, Y . (ii) There exist two functions Fi : R2 × Ω → R, i = 1, 2, which are B(R2 ) ⊗ Fi -measurable for i = 1, 2, such that |1 − g1 (Y )g2 (X) | = F1 (X, Y )F F2 (X, Y )

a.s.

Proof: Let G1 and G2 be two bounded nonnegative random variables such that Gi is Fi -measurable for i = 1, 2. Suppose that f : R2 −→ R is a nonnegative continuous and bounded function. For any x ∈ R we will denote by fi (x, ·) the density of the law of gi (x), for i = 1, 2. For each 1 1[−ε,ε] (z). Set ε > 0, deﬁne ϕε (z) = 2ε J(x, y) = |1 − g1 (y)g2 (x) |

−1

.

We will ﬁrst show the equality E [G1 G2 J(X, Y )f (X, Y )] (4.43) = E [G1 |g1 (y) = x ] f1 (y, x)E [G2 |g2 (x) = y ] f2 (x, y)f (x, y)dxdy. R2

Actually, we will see that both members arise when we compute the limit of E [G1 G2 ϕε (x − g1 (y))ϕε (y − g2 (x))] f (x, y)dxdy (4.44) R2

as ε tends to zero in two diﬀerent ways. For any ω ∈ Ω we introduce the mapping Φω : R2 → R2 deﬁned by Φω (x, y) = (x − g1 (y, ω), y − g2 (x, ω)) = (¯ x, y¯).

260

4. Transformations of the Wiener measure

Notice that Φω (X(ω), Y (ω)) = (0, 0). Denote by Dε0 (ω) the set : 9 Dε0 (ω) = (x, y) ∈ R2 : |x − g1 (y, ω)| < ε0 , |y − g2 (x, ω)| < ε0 . Hypotheses H2 and H3 imply that for almost all ω the mapping Φω is a C 1 -diﬀeomorphism from Dε0 (ω) onto (−ε0 , ε0 )2 . Therefore, making the change of variable (¯ x, y¯) = Φω (x, y), we obtain for any ε < ε0 ϕε (x − g1 (y))ϕε (y − g2 (x))f (x, y)dxdy R2 ϕε (¯ x)ϕε (¯)J(Φ−1 x, y¯))f (Φ−1 x, y¯))dxd ¯ y¯. = ω (¯ ω (¯ R2

By continuity this converges to J(X, Y )f (X, Y ) as ε tends to zero. The convergence of the expectations follows by the dominated convergence theorem, because from hypothesis H3 we have x, y¯)) ≤ J(Φ−1 ω (¯

sup |y−g2 (x,ω)|<ε0 |x−g1 (y,ω)|<ε0

|1 − g1 (y)g2 (x)|

−1

≤ H ∈ L1 (Ω)

if |¯ x| < ε0 , and |¯ y | < ε0 . Consequently, (4.44) converges to the left-hand side of (4.43) as ε tends to zero. Let us now turn to the proof that the limit of (4.44) equals the right-hand side of (4.43). We can write [G1 G2 ϕε (x − g1 (y))ϕε (y − g2 (x))] = E [G1 ϕε (x − g1 (y))] E [G2 ϕε (y − g2 (x))] ε = ϕ (x − α)E [G1 |g1 (y) = α ] f1 (y, α)dα R ε × ϕ (y − β)E [G2 |g2 (x) = β ] f2 (x, β)dβ . R

We are going to take the limit of both factors as ε tends to zero. For the ﬁrst one, the Lebesgue diﬀerentiation theorem tell us that for any y ∈ R there exists a set N y of zero Lebesgue measure such that for all x ∈ N y , lim ϕε (x − α)E [G1 |g1 (y) = α ] f1 (y, α)dα = E [G1 |g1 (y) = x ] f1 (y, x). ε↓0

R

In the same way, for the second integral, for each ﬁxed x ∈ R, there will be a set N x of zero Lebesgue measure such that for all y ∈ N x , lim ϕε (y − β)E [G2 |g2 (x) = β ] f2 (x, β)dβ = E [G2 |g2 (x) = y ] f2 (x, y). ε↓0

R

We conclude that, except on the set N = {(x, y) : x ∈ N y or y ∈ N x }

4.2 Markov random ﬁelds

261

we will have the convergence lim E [G1 G2 ϕε (x − g1 (y))ϕε (y − g2 (x))] ε↓0

= E [G1 |g1 (y) = x ] f1 (y, x)E [G2 |g2 (x) = y ] f2 (x, y). Thus, this convergence holds almost everywhere. The preceding equality provides the pointwise convergence of the integrands appearing in expression (4.44). The corresponding convergence of the integral is derived through the dominated convergence theorem, using hypothesis H1. Consequently, (4.43) holds for any continuous and bounded function f , and this equality easily extends to any measurable and bounded function f . Taking f = 1B , where B is a set of zero Lebesgue measure, and putting G1 = G2 = 1, we deduce from (4.43) that P {(X, Y ) ∈ B} = 0 because J(X, Y ) > 0 a.s. As a consequence, the law of (X, Y ) is absolutely continuous with a density given by fXY (x, y) =

f1 (x, y)ff2 (y, x) . E [J(X, Y ) |X = x, Y = y ]

Therefore, (4.43) implies that E [G1 G2 J(X, Y ) |X = x, Y = y ] fXY (x, y) = E [G1 |g1 (y) = x ] f1 (y, x)E [G2 |g2 (x) = y ] f2 (x, y),

(4.45)

almost surely with respect to the law of (X, Y ). Putting G2 = 1, we obtain E [G1 J(X, Y ) |X = x, Y = y ] fXY (x, y) = E [G1 |g1 (y) = x ] f1 (y, x)ff2 (x, y),

(4.46)

and with G1 ≡ 1 we get E [G2 J(X, Y ) |X = x, Y = y ] fXY (x, y) = E [G2 |g2 (x) = y ] f1 (y, x)ff2 (x, y).

(4.47)

Substituting (4.46) and (4.47) into (4.45) yields E [G1 G2 J(X, Y ) |X = x, Y = y ] E [J(X, Y ) |X = x, Y = y ]

(4.48)

= E [G1 J(X, Y ) |X = x, Y = y ] E [G2 J(X, Y ) |X = x, Y = y ] . Conditioning ﬁrst by the bigger σ-ﬁelds σ(X, Y ) ∨ F1 and σ(X, Y ) ∨ F2 in the right-hand side of (4.48), we obtain E [G1 G2 J(X, Y ) |XY ] E [J(X, Y ) |XY ] = E [G1 E [J(X, Y ) |X, Y, F1 ] |X, Y ] ×E [G2 E [J(X, Y ) |X, Y, F2 ] |X, Y ] .

(4.49)

262

4. Transformations of the Wiener measure

Suppose ﬁrst that F1

X,Y

F2 . This allows us to write Eq. (4.49) as follows:

E [G1 G2 J(X, Y )E [J(X, Y ) |X, Y ] |X, Y ] = E [G1 G2 E [J(X, Y ) |X, Y, F1 ] E [J(X, Y ) |X, Y, F2 ] |X, Y ] . Taking the expectation of both members of the above equality, we obtain J(X, Y )−1 =

E [J(X, Y ) |X, Y ] . E [J(X, Y ) |X, Y, F1 ] E [J(X, Y ) |X, Y, F2 ]

This implies the desired factorization because any random variable that is σ(X, Y ) ∨ Fi -measurable (i = 1, 2) can be written as F (X(ω), Y (ω), ω) for some B(R2 ) ⊗ Fi -measurable function F : R2 × Ω → R. Conversely, suppose that (ii) holds. Then we have from (4.49) E [G1 G2 |X, Y ] = E [G1 G2 F1 (X, Y )F F2 (X, Y ) J(X, Y ) |X, Y ] =

E [G1 F1 (X, Y )J(X, Y ) |X, Y ] E [G2 F2 (X, Y ) J(X, Y ) |X, Y ] . E [J(X, Y ) |X, Y ]

Writing this equality for G1 ≡ 1, G2 ≡ 1, and for G1 ≡ G2 ≡ 1, we conclude that E [G1 G2 |X, Y ] = E [G1 |X, Y ] E [G2 |X, Y ] . Remarks: Some of the conditions appearing in the preceding hypotheses can be weakened or modiﬁed, and the conclusion of Theorem 4.2.5 will continue to hold. In particular, in hypothesis H3 we can replace H(ω) H2 (ω), with Hi (ω) Fi -measurable for i = 1, 2, and assume only by H1 (ω)H H2 (ω) < ∞ a.s. In H1 the local integrability of the function δ(x, y) H1 (ω)H holds if the densities f1 (y, z) and f2 (x, z) of g1 (y) and g2 (x) are locally bounded in R2 . If the variables X and Y are discrete, then the conditional independence F2 is always true (see Exercise 4.2.9). F1 X,Y

The following two lemmas allow us to reformulate the factorization property appearing in the preceding theorem. Lemma 4.2.5 Suppose that (A1 , B1 ) and (A2 , B2 ) are R2 -valued independent random variables such that A1 A2 = B1 B2 . Then either (i) A1 = 0 a.s. or A2 = 0 a.s.; or (ii) there is a constant k = 0 such that A1 = kB1 a.s. and A2 = k −1 B2 a.s.

4.2 Markov random ﬁelds

263

Proof: Suppose that (i) does not hold. Then P (A1 = 0) = 0 and P (A2 = 0) = 0. Without loss of generality we can assume that the underlying probability space is a product space (Ω, F, P ) = (Ω1 , F1 , P1 )×(Ω2 , F2 , P2 ). Let ω 2 be such that A2 (ω 2 ) = 0. Then condition (ii) follows from the relationship B2 (ω 2 ) B1 (ω 1 ). A1 (ω 1 ) = A2 (ω 2 ) Lemma 4.2.6 Consider two independent σ-ﬁelds F1 , F2 and two random variables G1 , G2 such that Gi is Fi -measurable for i = 1, 2. The following statements are equivalent: (a) There exist two random variables H1 and H2 such that Hi is Fi measurable, i = 1, 2, and 1 − G1 G2 = H1 H2 . (b) G1 or G2 is constant a.s. Proof: The fact that (b) implies (a) is obvious. Let us show that (a) implies (b). As before we can assume that the underlying probability space is a product space (Ω1 × Ω2 , F1 ⊗ F2 , P1 × P2 ). Property (a) implies that ˜ 1 (˜ ˜ 1 (˜ [G ω 1 ) − G1 (ω 1 )]G2 (ω 2 ) = [H1 (ω 1 ) − H ω 1 )]H H2 (ω 2 ), 1 and H 1 are independent copies of G1 and H1 on some space where G (Ω1 , F1 , P1 ). Lemma 4.2.5 applied to the above equality implies either 1 − G1 = 0 a.s.; or (PA) G2 = 0 a.s. or G H2 for some constant k = 0. (PB) G2 = kH Then (PA) leads to property (b) directly and (PB) implies that 1 = [H1 + kG1 ]H H2 . Again applying Lemma 4.2.5 to this identity yields that H2 and thus G2 are a.s. constant. Corollary 4.2.1 Under the hypotheses of Theorem 4.2.5, assume in addition that 1 − g1 (Y )g2 (X) has constant sign. Then conditions (i) and (ii) are equivalent to the following statement: (iii) One (or both) of the variables g1 (Y ) and g2 (X) is almost surely constant with respect to the conditional law given X, Y . As an application of the above criterion of conditional independence we are going to provide an alternative proof of Theorem 4.2.1 under slightly Wt , t ∈ [0, 1]} be a Brownian motion dediﬀerent hypotheses. Let W = {W ﬁned in the canonical probability space (Ω, F, P ).

264

4. Transformations of the Wiener measure

Theorem 4.2.6 Let f and ψ be functions of class C 1 such that |f | ≤ K and ψ ≤ 0. Consider the equation t Xt = X0 − 0 f (Xs )ds + Wt (4.50) X0 = ψ(X1 ). This equation has a unique solution X = {Xt , t ∈ [0, 1]} that is a Markov random ﬁeld if and only if one of the following conditions holds: (a) f (x) = ax + b, for some constants a, b ∈ R; (b) ψ ≡ 0. Remarks: 1. The fact that a unique solution exists is easy (see Exercise 4.2.7). 2. The case X1 constant (condition (c) of Theorem 4.2.1) is not included in the above formulation. Proof: We will only show that if X is a Markov random ﬁeld then one of conditions (a) or (b) holds. We will assume that X is a Markov random ﬁeld and ψ (x0 ) = 0 for some x0 ∈ R. Fix 0 < s < t ≤ 1. The Markov ﬁeld property implies σ{Xr , r ∈ [s, t]}

σ{Xr , r ∈ (s, t)}.

σ{Xs ,Xt }

(4.51)

From the deﬁnition of the conditional independence we deduce i Fs,t

e Fs,t ,

σ{Xs ,Xt }

(4.52)

where i Fs,t = σ{W Wr − Ws , s ≤ r ≤ t}, and e Fs,t = σ{W Wr , 0 ≤ r ≤ s; Wr − Wt , t ≤ r ≤ 1}.

Indeed, (4.51) implies (4.52) because i ⊂ σ{Xr , r ∈ [s, t]} Fs,t

and e ⊂ σ{Xr , r ∈ (s, t)}. Fs,t

Deﬁne

ϕs,t (y) = y −

t

f (ϕs,r (y))dr + Wt − Ws , s

for any s ≤ t and y ∈ R. Consider the random functions g1 , g2 : R × Ω → R given by g1 (y) = ϕs,t (y) (4.53) g2 (x) = ϕ0,s (ψ(ϕt,1 (x))).

4.2 Markov random ﬁelds

We have

265

Xt = g1 (Xs ) Xs = g2 (Xt ),

and we are going to apply Theorem 4.2.5 to the functions g1 and g2 . Let us ﬁrst verify that these functions satisfy conditions H1 through H3 of this theorem. Proof of H1: over,

We have that g1 (y), g2 (x) ∈ D1,p for all p ≥ 2, and, moreDr g1 (y) = e−

t r

f (ϕs,u (y))du

1[s,t] (r),

and Dr g2 (x) = e−

s r

f (ϕ0,u (ψ(ϕt,1 (x))))du

1[0,s] (r)

+ϕ0,s (ψ(ϕt,1 (x)))ψ (ϕt,1 (x))e−

1 r

f (ϕt,u (x))du

1[t,1] (r).

From these explicit expressions for the derivative of the functions g1 and g2 we can deduce the absolute continuity of the laws of these variables using the criteria established in Chapter 2. In fact, we have D(gi ))H > 0, i = 1, 2, and we can apply Theorem 2.1.3. Furthermore, we have 1 1 P (|x − g1 (y)| < ε) ≤ eK (t − s)− 2 2ε

(4.54)

and

1 1 P (|y − g2 (x)| < ε) ≤ eK s− 2 , (4.55) 2ε which imply the boundedness of the function δ(x, y) introduced in H1. In z 1 1 (r)dr. fact, let us check Eq. (4.54). Set h = 1[s,t] and ψ (z) = 2 [x−,x+] −∞ We have Dh (g1 (y)) ≥ e−K (t − s). The duality formula (1.42) implies Dh [ψ (g1 (y))] 1 P (|x − g1 (y)|) = E 2ε Dh (g1 (y)) E((W Wt − Ws )ψ (g1 (y))) E(Dh [ψ (g1 (y))]) = ≤ −K e (t − s) e−K (t − s) 1

≤ eK (t − s)− 2 , and (4.54) holds. Proof of H2:

We are going to show that for all ω ∈ Ω the transformation (x, y) −→ (x − g1 (y, ω), y − g2 (x, ω))

is bijective from R2 to R2 . Let (¯ x, y¯) ∈ R2 . Set x = x ¯ + ϕs,t (y). It suﬃces to show that the mapping Γ y −→ ϕ0,s ψ ϕt,1 (¯ x + ϕs,t (y)) + y¯

266

4. Transformations of the Wiener measure

has a unique ﬁxed point, and this follows from dΓ x + ϕs,t (y)) ψ ϕt,1 (¯ x + ϕs,t (y)) dy = ϕ0,s ψ ϕt,1 (¯ x + ϕs,t (y))ϕs,t (y) ≤ 0. ×ϕt,1 (¯ Proof of H3:

We have

1 − g1 (y)g2 (x) = 1 − ϕs,t (y)ϕ0,s ψ ϕt,1 (x) ψ ϕt,1 (x) ϕt,1 (x) ≥ 1. Note that g1 (Xs )

t

= exp(−

and g2 (Xt ) = ψ (X1 ) exp(−

f (Xr )dr)

s

f (Xr )dr). [0,s]∪[t,1]

In view of Corollary 4.2.1, the conditional independence (4.52) implies that one of the variables g1 (Xs ) and g2 (Xt ) is constant a.s. with respect to the conditional probability, given Xs and Xt . Namely, there exists a measurable function h : R2 → R such that either t (1) exp(− s f (Xr )dr) = h(Xs , Xt ), or (2) ψ (X1 ) exp(− [0,s]∪[t,1] f (Xr )dr) = h(Xs , Xt ). We will show that (1) implies that f is constant. Case (2) would be treated in a similar way. Suppose that there exist two points x1 , x2 ∈ R such that f (x1 ) < α < f (x2 ). Let C be the class of continuous functions y : [0, 1] → R such that y0 = ψ(y1 ). The stochastic process X takes values in C, which is a closed subset of C([0, 1]), and the topological support of the law of X is C. As a consequence, we can assume that (1) holds for any X ∈ C. For any 0 < 2δ < t − s we can ﬁnd two trajectories z1 , z2 , ∈ C, depending on δ such that: (i) zi (s) = zi (t) = 12 (x1 + x2 ). (ii) The functions z1 and z2 coincide on [0, 1] − (s, t), and they do not depend on δ on this set. (iii) zi (r) = xi for all i = 1, 2 and r ∈ (s + δ, t − δ). (iv) The functions zi are linear on the intervals (s, s + δ) and (t − δ, t). We have

t lim exp( f (zi (r))dr) = e(t−s)f (xi ) . δ↓0

s

4.2 Markov random ﬁelds

267

We can thus ﬁnd a δ small enough such that t t (t−s)α exp f (z1 (r))dr < e < exp f (z2 (r))dr . s

s

By continuity we can ﬁnd neighborhoods Vi of zi in C, i = 1, 2, such that t t (t−s)α exp f (x(r))dr < e < exp f (y(r))dr , s

s

for all x ∈ V1 and y ∈ V2 . Therefore, h(Xs , Xt ) > e−(t−s)α if X ∈ V1 , and h(Xs , Xt ) < e−(t−s)α if X ∈ V2 . This is contradictory because there is a γ > 0 such that when x runs over Vi , i = 1, 2, the point (x(s), x(t)) takes all possible values on some rectangle [ 12 (x1 + x2 ) − γ, 12 (x1 + x2 ) + γ]2 .

Exercises 4.2.1 Let F1 , F2 , G be three sub-σ-ﬁelds in a probability space such that F2 . Show the following properties: F1 G

(a)

F1 ∨ G

(b)

F1

(c)

F1

G1

H

G

F2 ∨ G.

F2

if

G ⊂ G1 ⊂ F1 .

F2

if

G ⊂ F1 ∩ F2 , and H is a σ-ﬁeld containing G of the

form H = H1 ∨ H2 , where H1 ⊂ F1 and H2 ⊂ F2 . 4.2.2 Let {Gn , n ≥ 1} be a sequence of σ-ﬁelds such that F1 n. Show that F1

G

Gn

F2 for each

F2 , where G = ∨n Gn if the sequence is increasing, and

G = ∩n Gn if it is decreasing. 4.2.3 Let X = {Xt , t ∈ [0, 1]} be a continuous Markov process. Show that it satisﬁes the Markov ﬁeld property. 4.2.4 Let W = {W Wt , t ∈ [0, 1]} be a Brownian motion, and let g : R → R W1 ) is a Markov random be a measurable function. Show that Xt = Wt +g(W ﬁeld. Assume that g(x) = ax + b. Show that in this case X is a Markov process if and only if a = −1 or a = 0. 4.2.5 For any 0 < < 1 consider the function f : R+ → R deﬁned by ⎧ if 0≤t<1− ⎨ 1−t if 1−≤t<2+ f (t) = ⎩ 2+t if 2 + ≤ t.

268

4. Transformations of the Wiener measure

Let X = {Xt , t ≥ 0} be a stochastic process such that P {Xt = f (t), ∀t ≥ 0} = 12 and P {Xt = −ff (t), ∀t ≥ 0} = 12 . Show that X is a Markov process but X = lim↓0 X does not have the Markov property. 4.2.6 Consider the stochastic diﬀerential equation t Xt = X0 − a 0 Xs ds − b + Wt X0 = g(X1 − X0 ), where a, b ∈ R. Suppose that the implicit equation x = g((e−a − 1)x + y) has a unique continuous solution x = ϕ(y). Show that the above equation admits a unique explicit solution that is a Markov random ﬁeld. 4.2.7 Show that Eq. (4.50) has a unique continuous solution which is a Markov random ﬁeld if f (x) = ax + b. 4.2.8 Check the estimates (4.54) and (4.55) integrating by parts on the Wiener space. 4.2.9 Let A, B be two independent discrete random variables taking values in some countable set S. Consider two measurable functions f, g : R × S → R, and suppose that the system x = f (y, a) y = g(x, b) has a unique solution for each (a, b) ∈ S such that P {A = a, B = b} > 0. Let X, Y be the random variables determined by the equations X = f (Y, A) Y = g(X, B). Show that A and B are conditionally independent given X, Y . 4.2.10 Suppose that f, g are two real-valued functions satisfying the following conditions: (i) f is of class C 1 , and there exist K > 0 and λ ∈ R such that −λ ≤ f (x) ≤ K for all x.

(ii) g is of class C 1 , and eλ |g (x)| ≤ |1 + g (x)| for all x and for some λ > λ. Show that Eq. (4.21) has a unique solution for each ω ∈ C0 ([0, 1]) ([250] and [97]). 4.2.11 Let Q << P be two probabilities in a measurable space (Ω, F), and set η = dQ dP . Show that for any nonnegative (or Q-integrable) random variable ξ and for any sub-σ-algebra G ⊂ F we have EQ (ξ|G) =

EP (ξη|G) . EP (η|G)

4.2 Markov random ﬁelds

269

4.2.12 Consider the equation in R2

dXt + f (Xt ) = dW Wt X11 = X02 = 0,

where f (x1 , x2 ) = (x1 − x2 , −ff2 (x1 )) and f2 is a twice continuously diﬀerentiable function such that 0 ≤ f2 (x) ≤ K for some positive constant K. Show that there exists a unique solution, which is a Markov random ﬁeld if and only if f2 ≡ 0 (cf. [250]). 4.2.13 Let G(x, y) be the Green function of −∆ on a bounded domain D of Rk , k = 2, 3. Let W = {W (A), A ∈ B(D)} be a Brownian measure on D. Deﬁne the random ﬁeld U0 (x) = D G(x, y)W (dy), x ∈ D. Show that ¨ continuous paths of order 1 − if the process {U U0 (x), x ∈ D} has Holder k = 2, and Holder ¨ continuous paths of order 38 − if k = 3, for any > 0. Hint: Write G(x, y) as the sum of a smooth function plus a function with a singularity of the form log |x − y| if k = 2 and |x − y|−1 if k = 3, and use Kolmogorov’s continuity criterion.

Notes and comments [4.1] Proposition 4.1.2 is a fundamental result on nonlinear transformations of the Wiener measure and was obtained by Girsanov in [121]. Absolute continuity of the Wiener measure under linear (resp. nonlinear) transformations was discussed by Cameron and Martin in [56] (resp. [57]). We refer to Liptser and Shiryayev [200] for a nice and complete presentation of the absolute continuity of the transformations of the Wiener measure under adapted shifts. The extension of Girsanov’s theorem to nonlinear transformations was discussed by Ramer [290] and Kusuoka [178] in the context of an abstract Wiener space. The notion of H-continuously diﬀerentiable random variable and the material of Sections 4.1.3 and 4.1.5 have been taken from Kusuoka’s paper [178]. The case of a contraction (i.e., DuH⊗H < 1) has been studied by Buckdahn in [48]. In that case, and assuming some additional assumptions, one can show that there exists a transformation A : Ω → Ω verifying A ◦ T = T ◦ A = Id a.s., and the random variable η(u) has the following expression (in the case of the classical Wiener space): η(u) =

1 d[P ◦ A−1 ] = exp − δ(u) − u2H dt dP 2 1 t − Ds ut (Dt (us (At )))(T Tt )dsdt , 0

0

270

4. Transformations of the Wiener measure

where {T Tt , 0 ≤ t ≤ 1} is the one-parameter family of transformations of Ω deﬁned by s∧t ur (ω)dr (T Tt ω)s = ω s + 0

and {At , 0 ≤ t ≤ 1} is the corresponding family of inverse transformations. ¨ ¨ In [338] Ustunel and Zakai proved Proposition 4.1.5 under the hypothesis DuL(H,H) < 1 a.s. Theorem 4.1.2 has been generalized in diﬀerent directions. On one hand, local versions of this theorem can be found in Kusuoka [179]. On the other ¨ ¨ hand, Ustunel and Zakai [337] discuss the case where the transformation T is not bijective (a multiplicity function must be introduced in this case) and u is locally H-continuously diﬀerentiable. The case of a one-parameter family of transformations on the classical Wiener space {T Tt , 0 ≤ t ≤ 1} deﬁned by the integral equations t∧s ur (T Tr ω)dr (T Tt ω)(s) = ω s + 0

has been studied by Buckdahn in [49]. Assuming that u ∈ L1,2 is such that 1 1 ut 2∞ dt + 0 Dut H 2∞ dt < ∞, Buckdahn has proved that for each 0 t ∈ [0, 1] there exists a transformation At : Ω → Ω such that Tt ◦ At = Tt = Id a.s., P ◦T Tt−1 << P , P ◦A−1 << P , and the density functions At ◦T t −1 are given by of P ◦ Tt and P ◦ A−1 t t −1 d[P ◦ At ] 1 t = exp − Mt = us (T Ts )dW Ws − us (T Ts )2 ds dP 2 0 0 t s − (Dr us )(T Ts )Ds [ur (T Tr )]drds , 0 0 t d[P ◦ Tt−1 ] 1 t = exp us (T Ts At )dW Ws − us (T Ts At )2 ds Lt = dP 2 0 0 t s − (Ds ur )(T Tr At )Dr [us (T Ts At )]drds . 0

0

The process {Lt , 0 ≤ t ≤ 1} satisﬁes the Skorohod linear stochastic diﬀert ential equation Lt = 1 + 0 us Ls dW Ws . This provides a generalization of the results for this type of equations presented in Chapter 2. ¨ ¨ Ustunel and Zakai (cf. [335]) have extended this result to processes u such that: 1 1 E exp(λu2r )dr < ∞ and Dut H 4∞ dt < ∞ 0

0

for some λ > 0. They use a general expression of the Radon-Nikodym derivative associated with smooth ﬂows of transformations of Ω (see Cruzeiro [71]).

4.2 Markov random ﬁelds

271

In [93] Enchev and Stroock extend the above result to the case where the process u is Lipschitz, that means, DuH⊗H ≤ c a.s. ¨ ¨ We refer to the monograph by Ustunel and Zakai (cf. [339]) for a complete analysis of transformations on the Wiener space and their induced measures. [4.2] We refer to Rozanov [296] for a detailed analysis of the notion of conditional independence. The study of the Markov property for solutions to stochastic diﬀerential equations with boundary conditions by means of the change of probability technique was ﬁrst done in [250]. Further applications to diﬀerent type of equations can be found in [80], [97], [98], [246], [251], and [252]. The study of the germ Markov ﬁeld property for solutions to stochastic partial diﬀerential equations driven by a white noise has been done in [81], [82], and [253]. The characterization of the conditional independence presented in Section 4.2.3 has been obtained in [5].

5 Fractional Brownian motion

The fractional Brownian motion is a self-similar centered Gaussian process with stationary increments and variance equals t2H , where H is a parameter in the interval (0, 1). For H = 12 this process is a classical Brownian motion. In this chapter we will present the application of the Malliavin Calculus to develop a stochastic calculus with respect to the fractional Brownian motion.

5.1 Deﬁnition, properties and construction of the fractional Brownian motion A centered Gaussian process B = {Bt , t ≥ 0} is called fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has the covariance function RH (t, s) = E(Bt Bs ) =

1 2H s + t2H − |t − s|2H . 2

(5.1)

Fractional Brownian motion has the property: For : 9 following self-similar any constant a > 0, the processes a−H Bat , t ≥ 0 and {Bt , t ≥ 0} have the same distribution. This property is an immediate consequence of the fact that the covariance function (5.1) is homogeneous of order 2H.

274

5. Fractional Brownian motion

From (5.1) we can deduce the following expression for the variance of the increment of the process in an interval [s, t]: E |Bt − Bs |2 = |t − s|2H . (5.2) This implies that fBm has stationary increments. By Kolmogorov’s continuity criterion and (5.2) we deduce that fBm has a version with continuous trajectories. Moreover, by Garsia-RodemichRumsey Lemma (see Lemma A.3.1), we can deduce the following modulus of continuity for the trajectories of fBm: For all ε > 0 and T > 0, there exists a nonnegative random variable Gε,T such that E (|Gε,T |p ) < ∞ for all p ≥ 1, and |Bt − Bs | ≤ Gε,T |t − s|H−ε , for all s, t ∈ [0, T ]. In other words, the parameter H controls the regularity of the trajectories, which are H¨ o¨lder continuous of order H − ε, for any ε > 0. For H = 12 , the covariance can be written as R 12 (t, s) = t ∧ s, and the process B is a standard Brownian motion. Hence, in this case the increments of the process in disjoint intervals are independent. However, for H = 12 , the increments are not independent. Set Xn = Bn − Bn−1 , n ≥ 1. Then {Xn , n ≥ 1} is a Gaussian stationary sequence with covariance function ρH (n) =

1 (n + 1)2H + (n − 1)2H − 2n2H . 2

This implies that two increments of the form Bk −Bk−1 and Bk+n −Bk+n−1 are positively correlated (i.e. ρH (n) > 0) if H > 12 and they are negatively correlated (i.e. ρH (n) < 0) if H < 12 . In the ﬁrst case the process presents an aggregation behaviour and this property can be used to describe cluster phenomena. In the second case it can be used to model sequences with intermittency. In the case H > 12 the stationary sequence Xn exhibits long range dependence , that is, ρH (n) lim =1 n→∞ H(2H − 1)n2H−2

∞ and, as a consequence, n=1 ρH (n) = ∞. In the case H < 12 we have ∞

|ρH (n)| < ∞.

n=1

5.1.1 Semimartingale property We have seen that for H = 12 fBm does not have independent increments. The following proposition asserts that it is not a semimartingale.

5.1 Properties and construction of the fractional Brownian motion

275

Proposition 5.1.1 The fBm is not a semimartingale for H = 12 . Proof:

For p > 0 set Yn,p = npH−1

n Bj/n − B(j−1)/n p . j=1

By the self-similar property of fBm, the sequence {Y Yn,p , n ≥ 1} has the same distribution as {Yn,p , n ≥ 1}, where Yn,p = n−1

n

p

|Bj − Bj−1 | .

j=1

The stationary sequence {Bj − Bj−1 , j ≥ 1} is mixing. Hence, by the Ergodic Theorem Yn,p converges almost surely and in L1 (Ω) to E (|B1 |p ) as n tends to inﬁnity. As a consequence, Yn,p converges in probability as n tends to inﬁnity to E (|B1 |p ). Therefore, Vn,p =

n Bj/n − B(j−1)/n p j=1

converges in probability to zero as n tends to inﬁnity if pH > 1, and to inﬁnity if pH < 1. Consider the following two cases: i) If H < 12 , we can choose p > 2 such that pH < 1, and we obtain that the p-variation of fBm (deﬁned as the limit in probability limn→∞ Vn,p ) is inﬁnite. Hence, the quadratic variation (p = 2) is also inﬁnite. ii) If H > 12 , we can choose p such that H1 < p < 2. Then the p-variation is zero, and, as a consequence, the quadratic variation is also zero. On the other hand, if we choose p such that 1 < p < H1 we deduce that the total variation is inﬁnite. Therefore, we have proved that for H = tion cannot be a semimartingale.

1 2

the fractional Brownian mo

In [65] Cheridito has introduced the notion of weak semimartingale as a stochastic process {Xt , t ≥ 0} such that for each T > 0, the set of random variables ⎧ n ⎨ fj (Xtj − Xtj−1 ), n ≥ 1, 0 ≤ t0 < · · · < tn ≤ T, ⎩ j=1 5 |ffj | ≤ 1, fj is FtXj−1 -measurable

276

5. Fractional Brownian motion

is bounded in L0 (Ω), where for each t ≥ 0, FtX is the σ-ﬁeld generated by the random variables {Xs , 0 ≤ s ≤ t}. It is important to remark that this σ-ﬁeld is not completed with the null sets. Then, in [65] it is proved that fBm is not a weak semimartingale if H = 12 . Let us mention the following surprising result also proved in [65]. Suppose that {Bt , t ≥ 0} is a fBm with Hurst parameter H ∈ (0, 1), and {W Wt , t ≥ 0} is an ordinary Brownian motion. Assume they are independent. Set Mt = Bt + Wt . Then {M Mt , t ≥ 0} is not a weak semimartingale if H ∈ (0, 12 ) ∪ ( 12 , 34 ], and it is a semimartingale, equivalent in law to Brownian motion on any ﬁnite time interval [0, T ], if H ∈ ( 34 , 1).

5.1.2 Moving average representation Mandelbrot and Van Ness obtained in [217] the following integral representation of fBm in terms of a Wiener process on the whole real line (see also Samorodnitsky and Taqqu [301]). Proposition 5.1.2 Let {W (A), A ∈ B(R), µ(A) < ∞} be a white noise on R. Then H− 12 H− 12 1 Bt = (t − s)+ dW Ws , − (−s)+ C1 (H) R is a fractional Brownian motion with Hurst parameter H, if

∞

H− 12

(1 + s)

C1 (H) = 0

H− 1

H− 12

−s

2

1 ds + 2H

12 .

H− 1

Proof: Set ft (s) = ((t − s)+ ) 2 − ((−s)+ ) 2 , s ∈ R, t ≥ 0. Notice that R ft (s)2 ds < ∞. In fact, if H = 12 , as s tends to −∞, ft (s) behaves 3 as (−s)H− 2 which is square integrable at inﬁnity. For t ≥ 0 set H− 12 H− 12 (t − s)+ Xt = − (−s)+ dW Ws . R

We have E(Xt2 )

H− 12 H− 12 2 (t − s)+ − (−s)+ ds R H− 12 H− 12 2 (1 − u)+ = t2H − (−u)+ du R 1 0 2 H− 1 H− 1 = t2H (1 − u) 2 − (−u) 2 du + (1 − u)2H−1 du =

−∞

= C1 (H)2 t2H .

0

(5.3)

5.1 Properties and construction of the fractional Brownian motion

277

Similarly, for any s < t we obtain H− 12 H− 12 2 2 (t − u)+ E(|Xt − Xs | ) = − (s − u)+ du R H− 12 H− 12 2 (t − s − u)+ = − (−u)+ du R

= C1 (H)2 |t − s|2H .

(5.4)

From (5.3) and (5.4) we deduce that the centered Gaussian process {Xt , t ≥ 0} has the covariance RH of a fBm with Hurst parameter H. Notice that the above integral representation implies that the function RH deﬁned in (5.1) is a covariance function, that is, it is symmetric and nonnegative deﬁnite. It is also possible to establish the following spectral representation of fBm (see Samorodnitsky and Taqqu [301]): its e − 1 1 −H D 1 |s| 2 d Ws , Bt = C2 (H) R is D = W 1 + iW 2 is a complex Gaussian measure on R such that where W 1 W (A) = W 1 (−A), W 2 (A) = −W 2 (A), and E(W 1 (A)2 ) = E(W 2 (A)2 ) = |A| 2 , and 12 π . C2 (H) = HΓ(2H) sin Hπ

5.1.3 Representation of fBm on an interval Fix a time interval [0, T ]. Consider a fBm {Bt , t ∈ [0, T ]} with Hurst parameter H ∈ (0, 1). We denote by E the set of step functions on [0, T ]. Let H be the Hilbert space deﬁned as the closure of E with respect to the scalar product $ # 1[0,t] , 1[0,s] H = RH (t, s). The mapping 1[0,t] −→ Bt can be extended to an isometry between H and the Gaussian space H1 associated with B. We will denote this isometry by ϕ −→ B(ϕ). Then {B(ϕ), ϕ ∈ H} is an isonormal Gaussian process associated with the Hilbert space H in the sense of Deﬁnition 1.1.1. In this subsection we will establish the representation of fBm as a Volterra process using some computations inspired in the works [10] (case H > 12 ) and [240] (general case). Case H >

1 2

It is easy to see that the covariance of fBm can be written as t s RH (t, s) = αH |r − u|2H−2 dudr, 0

0

(5.5)

278

5. Fractional Brownian motion

where αH = H(2H − 1). Formula (5.5) implies that ϕ, ψH = αH

T

T

|r − u|2H−2 ϕr ψ u dudr 0

(5.6)

0

for any pair of step functions ϕ and ψ in E. We can write 1

|r − u|2H−2

=

(ru)H− 2 β(2 − 2H, H − 12 ) r∧u 3 3 × v 1−2H (r − v)H− 2 (u − v)H− 2 dv,

(5.7)

0

where β denotes the Beta function. Let us show Equation (5.7). Suppose r−v r and x = uz , we obtain r > u. By means of the change of variables z = u−v

=

u

3

r u

=

3

v 1−2H (r − v)H− 2 (u − v)H− 2 dv 0 ∞ 1−2H H− 32 (r − u)2H−2 (zu − r) z dz 1

(ru) 2 −H (r − u)2H−2

1

1−2H

(1 − x)

3

xH− 2 dx

0 1 1 = β(2 − 2H, H − )(ru) 2 −H (r − u)2H−2 . 2

Consider the square integrable kernel t 1 3 1 −H 2 (u − s)H− 2 uH− 2 du, KH (t, s) = cH s

(5.8)

s

where cH =

H(2H−1) β(2−2H,H− 12 )

1/2

and t > s.

Taking into account formulas (5.5) and (5.7) we deduce that this kernel veriﬁes t∧s t t∧s 3 1 KH (t, u)KH (s, u)du = c2H (y − u)H− 2 y H− 2 dy 0 0 u s H− 32 H− 12 1−2H × (z − u) z dz u du u

1 = − 2H, H − ) 2 = RH (t, s).

t

s

|y − z|2H−2 dzdy

c2H β(2

0

0

(5.9)

Formula (5.9) implies that the kernel RH is nonnegative deﬁnite and provides an explicit representation for its square root as an operator.

5.1 Properties and construction of the fractional Brownian motion

279

From (5.8) we get ∂KH (t, s) = cH ∂t

H− 12 3 t (t − s)H− 2 . s

(5.10)

∗ Consider the linear operator KH from E to L2 ([0, T ]) deﬁned by T ∂KH ∗ (t, s)dt. (5.11) ϕ)(s) = ϕ(t) (KH ∂t s

Notice that

∗ KH 1[0,t] (s) = KH (t, s)1[0,t] (s). ∗ KH

(5.12)

The operator is an isometry between E and L ([0, T ]) that can be extended to the Hilbert space H. In fact, for any s, t ∈ [0, T ] we have using (5.12) and (5.9) $ $ # # ∗ ∗ 1[0,s] L2 ([0,T ]) = KH (t, ·)1[0,t] , KH (s, ·)1[0,s] L2 ([0,T ]) KH 1[0,t] , KH t∧s = KH (t, u)KH (s, u)du 0 $ # = RH (t, s) = 1[0,t] , 1[0,s] H . 2

∗ The operator KH can be expressed in terms of fractional integrals: 1 1 1 H− 1 ∗ ϕ) (s) = cH Γ(H − )s 2 −H (IIT − 2 uH− 2 ϕ(u))(s). (KH 2

(5.13)

This is an immediate consequence of formulas (5.10), (5.11) and (A.14). For any a ∈ [0, T ], the indicator function 1[0,a] belongs to the image of ∗ and applying the rules of the fractional calculus yields (Exercise 5.1.6) KH 1 1 1 H− 1 ∗ −1 2 −H ) (1[0,a] ) = (KH Da− 2 uH− 2 (s)1[0,a] (s). (5.14) 1 s cH Γ(H − 2 ) Consider the process W = {W Wt , t ∈ [0, T ]} deﬁned by ∗ Wt = B((KH )

−1

(1[0,t] )).

(5.15)

Then W is a Wiener process, and the process B has the integral representation t Bt = KH (t, s)dW Ws . (5.16) 0

Indeed, for any s, t ∈ [0, T ] we have ∗ −1 ∗ −1 E(W Wt Ws ) = E B((KH ) (1[0,t] ))B((KH ) (1[0,s] )) A @ ∗ −1 ∗ −1 ) (1[0,t] ), (KH ) (1[0,s] ) = (KH H $ # = 1[0,t] , 1[0,s] L2 ([0,T ]) = s ∧ t.

280

5. Fractional Brownian motion

Moreover, for any ϕ ∈ H we have T ∗ (KH ϕ) (t)dW Wt . B(ϕ) =

(5.17)

0

Notice that from (5.14), the Wiener process W is adapted to the ﬁltration generated by the fBm B and (5.15) and (5.16) imply that both processes generate the same ﬁltration. Furthermore, the Wiener process W that provides the integral representation (5.16) is unique. Indeed, this follows from ∗ is L2 ([0, T ]), because this image the fact that the image of the operator KH contains the indicator functions. The elements of the Hilbert space H may not be functions but distributions of negative order (see Pipiras and Taqqu [283], [284]). In fact, from (5.13) it follows that H coincides with the space of distributions f such 1

H− 1

1

that s 2 −H I0+ 2 (f (u)uH− 2 )(s) is a square integrable function. We can ﬁnd a linear space of functions contained in H in the following way. Let |H| be the linear space of measurable functions ϕ on [0, T ] such that T T 2 2H−2 |ϕr | |ϕu | |r − u| drdu < ∞. (5.18) ϕ|H| = αH 0

0

It is not diﬃcult to show that |H| is a Banach space with the norm · |H| and E is dense in |H|. On the other hand, it has been shown in [284] that the space |H| equipped with the inner product ϕ, ψH is not complete and it is isometric to a subspace of H. The following estimate has been proved in [222]. Lemma 5.1.1 Let H >

1 2

1

and ϕ ∈ L H ([0, T ]). Then

ϕ|H| ≤ bH ϕ

1

L H ([0,T ])

,

(5.19)

for some constant bH > 0. Proof:

Using H¨ o¨lder’s inequality with exponent q =

2

ϕ|H| ≤ αH

T

|ϕr | 0

1 H

H⎛ T dr ⎝ 0

1 H

in (5.18) we get 1 1−H

T

2H−2

|ϕu | |r − u|

du

⎞1−H drr⎠

.

0

The second factor in the above expression, up to a multiplicative con2H−1 1 norm of the left-sided fractional integral I0+ |ϕ|. stant, is equal to the 1−H Finally is suﬃces to apply the Hardy-Littlewood inequality (see [317, Theorem 1, p.119])

α

I0+ f q ≤ cα,q f Lp (0,∞) (5.20) L (0,∞) where 0 < α < 1, 1 < p < q < ∞ satisfy 1 1 values α = 2H − 1, q = 1−H and p = H .

1 q

=

1 p

− α, with the particular

5.1 Properties and construction of the fractional Brownian motion

281

As a consequence 1

L2 ([0, T ]) ⊂ L H ([0, T ]) ⊂ |H| ⊂ H. The inclusion L2 ([0, T ]) ⊂ |H| can be proved by a direct argument:

T

T

2H−2

|ϕr | |ϕu | |r − u| 0

T

T

≤

drdu

0

≤

2

2H−2

|ϕu | |r − u| 0 0 2H−1

T H−

1 2

T

drdu

2

|ϕu | du. 0

T This means that the Wiener-type integral 0 ϕ(t)dBt (which is equal to B(ϕ), by deﬁnition) can be deﬁned for functions ϕ ∈ |H|, and

T

T

ϕ(t)dBt = 0

Case H <

∗ (KH ϕ) (t)dW Wt .

(5.21)

0

1 2

To ﬁnd a square integrable kernel that satisﬁes (5.9) is more diﬃcult than in the case H > 12 . The following proposition provides the answer to this problem. Proposition 5.1.3 Let H < 12 . The kernel KH (t, s)

E where cH =

( 1 H− 2 1 t = cH (t − s)H− 2 s ) 1 1 1 −H t H− 3 −(H − )s 2 u 2 (u − s)H− 2 du , 2 s

2H (1−2H)β(1−2H,H+1/2) ,

RH (t, s) =

satisﬁes

t∧s

KH (t, u)KH (s, u)du.

(5.22)

0

In the references [78] and [284] Eq. (5.22) is proved using the analyticity of both members as functions of the parameter H. We will give here a direct proof using the ideas of [240]. Notice ﬁrst that ∂KH 1 (t, s) = cH (H − ) ∂t 2

H− 12 3 t (t − s)H− 2 . s

(5.23)

282

5. Fractional Brownian motion

Proof: Consider ﬁrst the diagonal case s = t. Set φ(s) = We have (

φ(s)

s 0

KH (s, u)2 du.

s

s ( )2H−1 (s − u)2H−1 du u 0 s 1 1 sH− 2 u1−2H (s − u)H− 2 −(2H − 1) 0 s 3 1 × v H− 2 (v − u)H− 2 dv du

= c2H

u

1 +(H − )2 2

s

u

s

1−2H

v

0

H− 32

H− 12

(v − u)

)

2 dv

du .

u

Making the change of variables u = sx in the ﬁrst integral and using Fubini’s theorem yields φ(s)

! = c2H s2H β(2 − 2H, 2H) s 1 3 v H− 2 −(2H − 1)sH− 2 0 v 1−2H H− 12 H− 12 × u (s − u) (v − u) du dv 0 s v w 1 1 1 +2(H − )2 u1−2H (v − u)H− 2 (w − u)H− 2 2 0 0 0 H− 32 H− 32 ×w v dudwdv .

Now we make the change of variable u = vx, v = sy for the second term and u = wx, w = vy for the third term and we obtain & 1 1 + ) φ(s) = c2H s2H β(2 − 2H, 2H) − (2H − 1)( 4H 2 ' 1 1 1−2H H− 12 H− 12 × x (1 − xy) (1 − x) dxdy 0

0

= s2H . Suppose now that s < t. Diﬀerentiating Equation (5.22) with respect to t, we are aimed to show that H(t2H−1 − (t − s)2H−1 ) = 0

s

∂KH (t, u)KH (s, u)du. ∂t

(5.24)

5.1 Properties and construction of the fractional Brownian motion

Set φ(t, s) = φ(t, s)

s

283

∂KH ∂t (t, u)KH (s, u)du.

Using (5.23) yields 1 H− 2 s s H− 12 3 1 t 1 = c2H (H − ) (t − u)H− 2 (s − u)H− 2 du 2 0 u u s H− 12 3 1 t 1 −c2H (H − )2 (t − u)H− 2 u 2 −H 2 u 0 s 3 H− 2 H− 12 × v (v − u) dv du. 0

u

Making the change of variables u = sx in the ﬁrst integral and u = vx in the second one we obtain 1 t H− 1 φ(t, s) = c2H (H − ) (ts) 2 γ( ) 2 s s 1 3 1 t −c2H (H − )2 tH− 2 v H− 2 γ( ) dv, 2 v 0 1 1−2H 3 1 where γ(y) = 0 x (y − x)H− 2 (1 − x)H− 2 dx for y > 1. Then, (5.24) is equivalent to ' & 1 2 s H− 3 t t 1 H− 1 2 2 2 γ( ) − (H − ) v γ( ) dv cH (H − )s 2 s 2 v 0 1

1

= H(tH− 2 − t 2 −H (t − s)2H−1 ).

(5.25)

Diﬀerentiating the left-hand side of equation (5.25) with respect to t yields & ' s 3 5 1 1 3 t t c2H (H − ) (H − )sH− 2 δ( ) − (H − )2 v H− 2 δ( ) dv 2 2 s 2 v 0 : = µ(t, s), (5.26) where, for y > 1, δ(y) =

1

5

1

x1−2H (y − x)H− 2 (1 − x)H− 2 dx.

0

By means of the change of variables z =

y(1−x) y−x

we obtain

1 1 δ(y) = β(2 − 2H, H + )y −H− 2 (y − 1)2H−2 . 2 Finally, substituting (5.27) into (5.26) yields

µ(t, s)

(5.27)

3 1 1 = c2H β(2 − 2H, H + )(H − )(H − ) 2 2 2 1 1 1 ×t−H− 2 s(t − s)2H−2 + t−H− 2 ((t − s)2H−1 − t2H−1 ) 2 = H(1 − 2H) 1 1 3 1 1 × t−H− 2 s(t − s)2H−2 + (t − s)2H−1 t−H− 2 − tH− 2 . 2 2

284

5. Fractional Brownian motion

This last expression coincides with the derivative with respect to t of the right-hand side of (5.25). This completes the proof of the equality (5.22). The kernel KH can also be expressed in terms of fractional derivatives: 1 1 1 1 2 −H H− 2 (s). (5.28) KH (t, s) = cH Γ(H + )s 2 −H Dt− u 2 ∗ Consider the linear operator KH from E to L2 ([0, T ]) deﬁned by T ∂KH ∗ (t, s)dt. (5.29) (KH ϕ)(s) = KH (T, s)ϕ(s) + (ϕ(t) − ϕ(s)) ∂r s

Notice that

∗ KH 1[0,t] (s) = KH (t, s)1[0,t] (s).

(5.30)

From (5.22) and (5.30) we deduce as in the case H > 12 that the operator ∗ is an isometry between E and L2 ([0, T ]) that can be extended to the KH Hilbert space H. ∗ can be expressed in terms of fractional derivatives: The operator KH 1

1

−H

1

∗ ϕ) (s) = dH s 2 −H (DT2 − uH− 2 ϕ(u))(s), (KH

(5.31)

where dH = cH Γ(H + 12 ). This is an immediate consequence of (5.29) and the equality 1 1 1 1 −H 2 −H H− 2 Dt− (s)1[0,t] (s) = DT2 − uH− 2 1[0,t] (u) (s). u As a consequence, C γ ([0, T ]) ⊂ H ⊂ L2 ([0, T ]), if γ > 12 − H. Using the alternative expression for the kernel KH given by 1 1 t KH (t, s) = cH (t − s)H− 2 + sH− 2 F1 ( ), s

where 1 F1 (z) = cH ( − H) 2

z−1

3

(5.32) 1

θH− 2 (1 − (θ + 1)H− 2 )dθ, 0

1 2 −H

one can show that H = IT − (L2 ) (see [78] and Proposition 6 of [9]). In 1

−H

fact, from (5.29) and (5.32) we obtain, for any function ϕ in IT2 − (L2 ) ∗ (KH ϕ) (s)

1

= cH (T − s)H− 2 ϕ(s) T 3 1 (ϕ(r) − ϕ(s))(r − s)H− 2 dr +cH (H − ) 2 s T r 3 +sH− 2 dr ϕ(r)F F1 s s 1 1 −H = cH Γ( + H)DT2 − ϕ(s) + Λϕ(s), 2

5.1 Properties and construction of the fractional Brownian motion

285

where the operator 1 Λϕ(s) = cH ( − H) 2

T

3

ϕ(r)(r − s)H− 2 s

r H− 12 1− dr s

2

is bounded in L . On the other hand, (5.31) implies that 1

1

−H

1

2 −H (I IT2 − uH− 2 φ(u))(s)}, H = {f : ∃φ ∈ L2 (0, T ) : f (s) = d−1 H s

with the inner product f, gH = if

T

φ(s)ψ(s)ds, 0 1

1

−H

1

2 −H (I IT2 − uH− 2 φ(u))(s) f (s) = d−1 H s

and

1

1

−H

1

2 −H (I IT2 − uH− 2 ψ(u))(s). g(s) = d−1 H s

Consider process W = {W Wt , t ∈ [0, T ]} deﬁned by ∗ ) Wt = B((KH

−1

(1[0,t] )).

As in the case H > 12 , we can show that W is a Wiener process, and the process B has the integral representation t Bt = KH (t, s)dW Ws . 0

Therefore, in this case the Wiener-type integral 1 2 −H

T 0

ϕ(t)dBt can be deﬁned

for functions ϕ ∈ IT − (L ), and (5.21) holds. 2

Remark In [9] these results have been generalized to Gaussian Volterra processes of the form t Xt = K(t, s)dW Ws , 0

where {W Wt , t ≥ 0} is a Wiener process and K(t, s) is a square integrable kernel. Two diﬀerent types of kernels can be considered, which correspond to the cases H < 12 and H > 12 : i) Singular case: K(·, s) has bounded variation on any interval (u, T ], T u > s, but s |K|(dt, s) = ∞ for every s. T ii) Regular case: The kernel satisﬁes s |K|((s, T ], s)2 ds < ∞ for each s.

286

5. Fractional Brownian motion

Deﬁne the left and right-sided fractional derivative operators on the whole real line for 0 < α < 1 by ∞ f (s) − f (s + u) α α f (s) := du D− Γ(1 − α) 0 u1+α ∞ α f (s) − f (s − u) := du, Γ(1 − α) 0 u1+α s ∈ R, respectively. Then, the scalar product in H has the following simple expression @ 1 A 1 −H −H f, gH = e2H D−2 f, D+2 g , (5.33) and

α f (s) D+

L2 (R)

−1

where eH = C1 (H) if s ∈ / [0, T ].

Γ(H+ 12 ),

f, g ∈ H, and by convention f (s) = g(s) = 0

Exercises 5.1.1 Show that B = {Bt , t ≥ 0} is a fBm with Hurst parameter H if and only if it is a centered Gaussian process with stationary increments and variance t2H . 5.1.2 Using the self-similarity property of the fBm show that for all p > 0 E sup |Bt |p = Cp,H T pH . 0≤t≤T

5.1.3 Let B = {Bt , t ≥ 0} be a fBm with Hurst parameter H ∈ (0, 12 ) ∪ ( 12 , 1). Show that the following process is a martingale t 1 1 Mt = s 2 −H (t − s) 2 −H dBs 0 2−2H

with variance c1,H t and compute the constant c1,H . Hint: Use the representation (5.17).

t 1 Ys , 5.1.4 Show that the fBm admits the representation Bt = c2,H 0 sH− 2 dY t 1 1 H− 2 H− 2 where Yt = 0 (t−s) s dW Ws , and W is an ordinary Brownian motion.

5.1.5 Suppose H > that for all p > 0

1 2.

If τ is a stopping time with values in [0, T ], show

E

sup |Bt |

p

0≤t≤τ

≤ Cp,H E(τ pH ).

Hint: Use the representation established in Exercise 5.1.4. 5.1.6 Show formula (5.14). 5.1.7 Show that |H| is a Banach space with the norm · |H| and E is dense in |H|. 5.1.8 Show formula (5.33).

5.2 Stochastic calculus with respect to fBm

287

5.2 Stochastic calculus with respect to fBm In this section we develop a stochastic calculus with respect to the fBm. There are essentially two diﬀerent approaches to construct stochastic integrals with respect to the fBm: (i) Path-wise approach. If u = {ut , t ∈ [0, T ]} is a stochastic process with γ-H¨ older continuous trajectories, where γ > 1−H, then by the results T of Young ([354]) the Riemann Stieltjes integral 0 ut dBt exists pathwise. This method is particularly useful in the case H > 12 , because it includes processes of the form ut = F (Bt ), where F is a continuously diﬀerentiable function. (ii) Malliavin calculus. We have seen in Chapter 1 that in the case of an ordinary Brownian motion, the adapted processes in L2 ([0, T ] × Ω) belong to the domain of the divergence operator, and on this set the divergence operator coincides with Itˆoˆ’s stochastic integral. Actually, the divergence operator coincides with an extension of Itˆ oˆ’s stochastic integral introduced by Skorohod in [315]. In this context a natural question is to ask in which sense the divergence operator with respect to a fractional Brownian motion B can be interpreted as a stochastic integral. Note that the divergence operator provides an isometry between the Hilbert Space H associated with the fBm B and the Gaussian space H1 , and gives rise to a notion of stochastic integral for classes of deterministic functions included in H. If H < 12 , 1

−H

then H = IT2 − (L2 ) is a class of functions that contains C γ ([0, T ]) if γ > 12 − H. If H = 12 , then H = L2 ([0, T ]), and if H > 12 , H contains the space |H| of functions. We will see that in the random case, (see Propositions 5.2.3 and 5.2.4) the divergence equals to a path-wise integral minus the trace of the derivative.

5.2.1

Malliavin Calculus with respect to the fBm

Let B = {Bt , t ∈ [0, T ]} be a fBm with Hurst parameter H ∈ (0, 1). The process {B(ϕ), ϕ ∈ H} is an isonormal Gaussian process associated with the Hilbert space H in the sense of Deﬁnition 1.1.1. We will denote by D and δ the derivative and divergence operators associated with this process. ∗ is an isometry between H and a closed Recall that the operator KH 2 ∗ −1 ) (1[0,t] )) is a Wiener subspace of L ([0, T ]). Moreover, Wt = B((KH process such that t KH (t, s)dW Ws , Bt = 0 ∗ and for any ϕ ∈ H we have B(ϕ) = W (KH ϕ).

288

5. Fractional Brownian motion

In this framework there is a transfer principle that connects the derivative and divergence operators of both processes B and W . Its proof is left as an exercise (Exercise 5.2.1). 1,2 Proposition 5.2.1 For any F ∈ DW = D1,2 ∗ KH DF = DW F,

where DW denotes the derivative operator with respect to the process W , 1,2 the corresponding Sobolev space. and DW −1

∗ Proposition 5.2.2 Domδ = (KH ) (Domδ W ), and for any H-valued ran∗ u), where δ W denotes dom variable u in Dom δ we have δ(u) = δ W (KH the divergence operator with respect to the process W .

Suppose H > 12 . We denote by |H|⊗|H| the space of measurable functions ϕ on [0, T ]2 such that 2 ϕr,θ ϕu,η |r − u|2H−2 |θ − η|2H−2 drdudθdη < ∞. ϕ|H|⊗|H| = α2H [0,T ]4

Then, |H| ⊗ |H| is a Banach space with respect to the norm · |H|⊗|H| . Furthermore, equipped with the inner product 2H−2 2H−2 ϕr,θ ψ u,η |r − u| |θ − η| drdudθdη ϕ, ψH⊗H = α2H [0,T ]4

the space |H| ⊗ |H| is isometric to a subspace of H ⊗ H. A slight extension of the inequality (5.19) yields ϕ|H|⊗|H| ≤ bH ϕ

1

L H ([0,T ]2 )

.

(5.34)

For any p > 1 we denote by D1,p (|H|) the subspace of D1,p (H) formed by the elements u such that u ∈ |H| a.s., Du ∈ |H| ⊗ |H| a.s., and p p E u|H| + E Du|H|⊗|H| < ∞.

5.2.2

Stochastic calculus with respect to fBm. Case H >

1 2

We can introduce the Stratonovich integral as the limit of symmetric Riemann sums as we have done in Chapter 3 in the framework of the anticipating stochastic calculus for the Brownian motion. However, here we will consider only uniform partitions, because this simplify the proof of the results. T Consider a measurable process u = {ut , t ∈ [0, T ]} such that 0 |ut |dt < ∞ a.s. Let us deﬁne the aproximating sequences of processes ti+1 n−1 (π n u)t = ∆−1 u ds 1(ti ,ti+1 ] (t), (5.35) s n i=0

ti

5.2 Stochastic calculus with respect to fBm T n.

where ti = i∆n , i = 0, . . . , n, and ∆n = n−1

n

S =

∆−1 n

Set

ti+1

us ds (Bti+1 − Bti ).

ti

i=0

289

Deﬁnition 5.2.1 We say that a measurable process u = {ut , 0 ≤ t ≤ T } T such that 0 |ut |dt < ∞ a.s. is Stratonovich integrable with respect to the fBm if the sequence S n converges in probability as |π| → 0, and in this case T the limit will be denoted by 0 ut ◦ dBt . The following proposition establishes the relationship between the Stratonovich integral and the divergence integral. It is the counterpart of Theorem 3.1.1 for the fBm. Proposition 5.2.3 Let u = {ut , t ∈ [0, T ]} be a stochastic process in the space D1,2 (|H|). Suppose also that a.s.

T

T

2H−2

|Ds ut | |t − s| 0

dsdt < ∞.

(5.36)

0

Then u is Stratonovich integrable and we have

T

T

ut ◦ dBt = δ(u) + αH 0

T

2H−2

Ds ut |t − s| 0

dsdt.

(5.37)

0

Proof: The proof will be done in two steps. T Step 1. Notice ﬁrst that 0 |ut |dt < ∞ a.s. because |H| ⊂ L1 ([0, T ]). We claim that 2 2 (5.38) π n u|H| ≤ dH u|H| , for some positive constant dH . In fact, we have that π

n

2 u|H|

T

T

|(π n u)s | |(π n u)t | |s − t|2H−2 dsdt

= αH 0

0

T

≤ αH

T

|us | |ut | φn (s, t)dsdt, 0

0

where φn (s, t) = ∆−2 n

n−1 i,j=0

1(ti ,ti+1 ] (s)1(tj ,tj+1 ] (t)

ti+1

ti

tj+1

tj

Hence, in order to show (5.38) it suﬃces to check that φn (s, t)|s − t|2−2H ≤ dH .

|σ − θ|2H−2 dσdθ.

290

5. Fractional Brownian motion

Notice that ti+1 αH ti

tj+1

tj

|σ − θ|2H−2 dσdθ = E((Bti+1 − Bti )(Btj+1 − Btj )).

Thus, for s, t ∈ (ti , ti+1 ] −1 2H 2−2H ≤ α−1 φn (s, t)|s − t|2−2H = ∆−2 n αH ∆n |s − t| H ,

and for s ∈ (ti , ti+1 ], t ∈ (tj , tj+1 ] with i < j

=

≤

φn (s, t)|s − t|2−2H |s − t|2−2H ! (tj − ti+1 )2H + (tj − ti+1 + 2∆n )2H 2αH ∆2n " −2(tj − ti+1 + ∆n )2H ! " 1 max k 2H + (k + 2)2H − 2(k + 1)2H (k + 2)2−2H . 2αH k≥0

Therefore (5.38) holds with dH = α−1 H max(1, k≥0

" 1 ! 2H k + (k + 2)2H − 2(k + 1)2H (k + 2)2−2H ). 2 2

We can ﬁnd a sequence of step processes uk such that uk − u|H| → 0 as k tends to inﬁnity. Then π n u − u|H|

≤

π n u − π n uk |H| + π n uk − uk |H| + uk − u|H| % ≤ ( dH + 1) uk − u|H| + π n uk − uk |H| ,

and letting ﬁrst n tend to inﬁnity and then k tend to infnity we get lim π n u − u|H| = 0

n→∞

a.s., and by dominated convergence we obtain 2

lim E(π n u − u|H| ) = 0.

n→∞

In a similar way we can show that 2

2

Dπ n u|H|⊗|H| ≤ fH Du|H|⊗|H| , for some constant fH > 0, and as a consequence 2

lim E(Dπ n u − Du|H|⊗|H| ) = 0.

n→∞

Therefore, π n u → u in the norm of the space D1,2 (|H|).

5.2 Stochastic calculus with respect to fBm

291

Step 2. Using Proposition 1.3.3 we can write n−1

∆−1 n

ti+1

ti

i=0

where Tn (u) =

us ds (Bti+1 − Bti ) = δ(π n u) + Tn (u),

n−1

∆−1 n

ti+1

ti

i=0

(5.39)

# $ Dus , 1[ti ,ti+1 ] H ds.

By Step 1 δ(π n u) will converge in L2 (Ω) to δ(u) as n tends to inﬁnity. Then it suﬃces to show that Tn (u) converges almost surely to

T

T

2H−2

Ds ut |t − s|

αH 0

We can write

T

Tn (u) = αH

ψ n (s, t) =

n−1

T

Ds ut ψ n (s, t)dsdt, 0

where

dsdt.

0

0

1[ti ,ti+1 ] (t)∆−1 n

i=0

ti+1

|s − σ|2H−2 dσ.

ti

By dominated convergence it suﬃces to show that ψ n (s, t)|s − t|2−2H ≤ eH for some constant eH > 0. If s, t ∈ (ti , ti+1 ] then, ψ n (s, t)|s − t|2−2H

2−2H ≤ ∆−1 (2H − 1)−1 n ∆n " ! × (ti+1 − s)2H−1 + (s − ti )2H−1 2 . ≤ 2H − 1

On the other hand, if s ∈ (ti , ti+1 ], t ∈ (tj , tj+1 ] with i < j we have ψ n (s, t)|s − t|2−2H

≤

1 sup (k + 2)2−2H 2H − 1 k≥0 λ∈[0,1]

×[(k + 1 + λ)2H−1 − (k + λ)2H−1 ] gH . = 2H − 1 Hence, (5.2.2) holds with eH = of the proposition.

1 2H−1

max(2, gH ). This completes the proof

292

5. Fractional Brownian motion

Remark 1 A suﬃcient condition for (5.36) is 1/p T

T

p

|Ds ut | dt 0

for some p >

ds < ∞

s

1 2H−1 .

Remark 2 Let u = {ut , t ∈ [0, T ]} be a stochastic process which is continuous in the norm of D1,2 and (5.36) holds. Then the Riemann sums n−1

usi (Bti+1 − Bti ),

i=0

where ti ≤ si ≤ ti+1 , converge in probability to the right-hand side of (5.37). In particulat, the forward and backward integrals of u with respect to the fBm exists and they coincide with the Stratonovich integral. (A) The divergence integral Suppose that u = {ut , t ∈ [0, T ]} is a stochastic process in the space D1,2 (|H|). Then, for any t ∈ [0, T ] the process u1[0,t] also belongs to D1,2 (|H|) and we can deﬁne the indefnite divergence integral denoted by t us dBs = δ u1[0,t] . 0

If (5.36) holds, then by Proposition 5.2.3 we have t t T t 2H−2 us ◦ dBs = us dBs + αH Dr us |s − r| drds. 0

0

0

0

By Proposition 1.5.8, if p > 1, a process u ∈ D1,p (|H|) belongs to the domain of the divergence in Lp (Ω), and we have p p p E (|δ (u)| ) ≤ CH,p E (u)|H| + E Du|H|⊗|H| . As a consequence, applying (5.34) we obtain p p p E (|δ (u)| ) ≤ CH,p E (u)L1/H ([0,T ]) + E DuL1/H ([0,T ]2 ) . (5.40) 1,2 Let pH > 1. Denote by L1,p (|H|) such H the space of processes u ∈ D that ⎡ ⎛ pH ⎞⎤ p1 T T T 1 up,1 := ⎣ E(|us |p )ds + E ⎝ |Dr us | H dr ds⎠⎦ < ∞. 0

0

0

Assume pH > 1 and suppose that u ∈ L1,p H and consider the indeﬁnite t divergence integral Xt = 0 us dBs . The following results have been established in [6]:

5.2 Stochastic calculus with respect to fBm

293

(i) Maximal inequality for the divergence integral: E

p

sup |Xt | t∈[0,T ]

p

≤ C up,1 ,

where the constant C > 0 depends on p, H and T . This follows from the Lp estimate (5.40) and a convolution argument. (ii) Continuity: The process Xt has a version with continuous trajectories and for all γ < H − p1 there exists a random variable Cγ such that γ

|Xt − Xs | ≤ Cγ |t − s| . 1,p As a consequence, 4 for a process u ∈5∩p>1 L H , the indeﬁnite integral t u dBs , t ∈ [0, T ] is γ-Holder ¨ continuous for all process X = 0 s γ < H.

(B) Ito’s ˆ formula for the divergence integral Suppose that f, g : [0, T ] −→ R are H¨¨older continuous functions of orders α and β with α + β > 1. Young [354] proved that the Riemann-Stieltjes t T integral 0 fs dgs exists. Moreover, if ht = 0 fs dgs and F is of class C 2 the following change of variables formula holds: t F (ht ) = F (0) + F (hs )ffs dgs . 0

As a consequence, if F is a function of class C 2 , and H > 12 , the t Stratonovich integral integral 0 F (Bs )◦dBs introduced in Deﬁnition 5.2.1 is actually a path-wise Riemann-Stieltjes integral and for any function F of class C 2 we have t F (Bs ) ◦ dBs . (5.41) F (Bt ) = F (0) + 0

Suppose that F is a function of class C 2 (R) such that 2

max {|F (x)|, |F (x)|, |F (x)|} ≤ ceλx ,

(5.42)

where c and λ are positive constants such that λ < 4T12H . This condition implies 2 E sup |F (Bt )|p ≤ cp E epλ sup0≤t≤T |Bt | < ∞ 0≤t≤T

−2H

for all p < T 2λ . In particular, we can take p = 2. The same property holds for F and F .

294

5. Fractional Brownian motion

Then, if F satisﬁes the growth condition (5.42), the process F (Bt ) belongs to the space D1,2 (|H|) and (5.36) holds. As a consequence, from Proposition 5.2.3 we obtain

t

t s F (Bs )dBs + αH F (Bs )(s − r)2H−2 drds 0 0 0 t t F (Bs )dBs + H F (Bs )s2H−1 ds. (5.43) =

F (Bs ) ◦ dBs

t

=

0

0

0

Therefore, putting together (5.41) and (5.43) we deduce the following Itˆ oˆ’s formula for the divergence process F (Bt ) = F (0) +

t

F (Bs )dBs + H

0

t

F (Bs )s2H−1 ds.

(5.44)

0

We recall that the divergence operator has the local property and δ(u) 1,2 (H). is deﬁned without ambiguity in Dloc We state the following general version of Itˆ o’s formula (see [11]). Theorem 5.2.1 Let F be a function of class C 2 (R). Assume that u = 2,2 (|H|) such that the indeﬁnite {ut , t ∈ [0, T ]} is a process in the space Dloc t integral Xt = 0 us dBs is a.s. continuous. Assume that u2 belongs to H. Then for each t ∈ [0, T ] the following formula holds F (Xt ) = F (0) +

0 t

+ αH

t

F (Xs )us dBs T

|s − σ|

F (Xs ) us 0

+ αH

t

F (Xs )us

2H−2

Dσ uθ dBθ dσ ds

0

0

s

uθ (s − θ)

0

s

2H−2

dθ ds.

(5.45)

0

Remark 1 If the process u is adapted, then the third summand in the right-hand side of (5.45) can be written as t

αH 0

s

F (Xs ) us

θ

|s − σ| 0

2H−2

Dσ uθ dσ dBθ

ds.

0

2H−2 Remark 2 s2H−1 1[0,s] (θ) is an approximation of the identity as 2H−1 (s − θ) 1 H tends to 2 . Therefore, taking the limit as H converges to 12 in Equation (5.45) we recover the usual Itˆ o’s formula for the the Skorohod integral (see Theorem 3.2.2).

5.2 Stochastic calculus with respect to fBm

295

5.2.3 Stochastic integration with respect to fBm in the case H < 12 The extension of the previous results to the case H < 12 is not trivial and new diﬃculties appear. In order to illustrate these diﬃculties, let us ﬁrst T remark that the forward integral 0 Bt dBt deﬁned as the limit in L2 of the Riemann sums n−1 Bti (Bti+1 − Bti ), i=0

where ti = iT n , does not exists. In fact, a simple argument shows that the expectation of this sum diverges: n

E Bti−1 (Bti − Bti−1 ) =

i=1

=

" 1 ! 2H 2H ti − t2H i−1 − (ti − ti−1 ) 2 i=1 n

1 2H 1 − n1−2H → −∞, T 2

as n tends to inﬁnity. Notice, however, that the expectation of symmetric Riemann sums is constant: n n ! 2H " T 2H 1 1 ti − t2H . E (Bti + Bti−1 )(Bti − Bti−1 ) = i−1 = 2 i=1 2 i=1 2 ∗ We recall that for H < 12 the operator KH given by (5.31) is an isometry 2 between the Hilbert space H and L ([0, T ]). We have the estimate : ∂K 1 H− 32 . (5.46) ∂t (t, s) ≤ cH ( 2 − H) (t − s)

Also from (5.32) (see Exercise 5.2.3) it follow sthat 1

|K(t, s)| ≤ C(t − s)H− 2 .

(5.47)

Consider the following seminorm on the set E of step functions on [0, T ]: 2

ϕK

T

ϕ2 (s)(T − s)2H−1 ds

= 0

T

T

2 H− 32

|ϕ(t) − ϕ(s)| (t − s)

+ 0

dt

ds.

s

We denote by HK the completion of E with respect to this seminorm. The 2 space HK is the class of functions ϕ on [0, T ] such that ϕK < ∞, and it 1,2 is continuously included in H. If u ∈ D (HK ), then u ∈ Dom δ.

296

5. Fractional Brownian motion

Note that if u = {ut , t ∈ [0, T ]} is a process in D1,2 (HK ), then there is a sequence {ϕn } of bounded simple HK -valued processes of the form ϕn =

n−1

Fj 1(tj ,tj+1 ] ,

(5.48)

j=0

where Fj is a smooth random variable of the form Fj = fj (Bsj , ..., Bsj 1

),

m(j)

with fj ∈ Cb∞ (Rm(j) ), and 0 = t0 < t1 < ... < tn = T , such that T 2 2 E u − ϕn K + E Dr u − Dr ϕn K dr −→ 0, as n → ∞. 0

(5.49) In the case H < 12 it is more convenient to consider the symmetric integral introduced by Russo and Vallois in [298]. For a process u = {ut , t ∈ [0, T ]} t+ε with integrable paths and ε > 0, we denote by uεt the integral (2ε)−1 t−ε us ds, / [0, T ]. Also we put Bs = BT where we use the convention us = 0 for s ∈ for s > T and Bs = 0 for s < 0. Deﬁnition 5.2.2 The symmetric integral of a process u with integrable paths with respect to the fBm is deﬁned as the limit in probability of T −1 us (Bs+ε − Bs−ε ) ds. (2ε) 0

as ε ↓ 0 if it exists. We denote this limit by

T 0

ur ◦ dBr .

The following result is the counterpart of Proposition 5.2.3 in the case H < 12 , for the symmetric integral. Proposition 5.2.4 Let u = {ut , t ∈ [0, T ]} be a stochastic process in the space D1,2 (HK ). Suppose that the trace deﬁned as the limit in probability T # $ 1 Dus , 1[s−ε,s+ε]∩[0,T ] H ds TrDu := lim ε→0 2ε 0 exists. Then the symmetric stochastic integral of u with respect to fBm in the sense of Deﬁnition 5.2.1 exists and T ut ◦ dBt = δ(u) + TrDu. 0

In order to prove this theorem, we need the following technical result. Lemma 5.2.1 Let u be a simple process of the form (5.48). Then uε converges to u in D1,2 (HK ) as ε ↓ 0.

5.2 Stochastic calculus with respect to fBm

297

Proof: Let u be given by the right-hand side of (5.48). Then u is a bounded process. Hence, by the dominated convergence theorem

T

(us − uεs )2 (T − s)2H−1 ds −→ 0

E

ε ↓ 0.

as

(5.50)

0

Fix an index i ∈ {0, 1, ..., n − 1}. Using that ut − us = 0 for s, t ∈ [ti, ti+1 ] we obtain

ti+1

2

T

|uεt

ti

−

− (ut − us )| (t − s)

dt

ds

s

ti+1

≤ 2

ti

ti+1

|uεt

−

uεs | (t

H− 32

− s)

2 dt

ds

s

ti+1

2

T

|uεt

+2 ti

=

H− 32

uεs

−

H− 32

− (ut − us )| (t − s)

uεs

dt

ds

ti+1

2A1 (i, ε) + 2A2 (i, ε).

(5.51)

The convergence of the expectation of the term A2 (i, ε) to 0, as ε ↓ 0, follows from the dominated convergence theorem, the fact that u is a bounded process and that for a.a. 0 ≤ s < t ≤ T, H− 32

|uεt − uεs − (ut − us )| (t − s)

−→ 0

ε ↓ 0.

as

Suppose that ε < 14 min0≤i≤n−1 |ti+1 − ti |. Then uεt − uεs = 0 if s and t belong to [ti + 2ε, ti+1 − 2ε], we can make the following decomposition E(A1 (i, ε)) ti +2ε ≤ 8 ti

ti +2ε

|uεt

ti+1

ti+1

ti+1 −2ε

− s)

2 dt

ds

ti +2ε

ti ti+1 −2ε

ti+1

2

3

ds

2

|uεt − uεs | (t − s)H− 2 dt

ds

ti +2ε

ti+1

+8 ti

3

|uεt − uεs | (t − s)H− 2 dt

s

+8

H− 32

s

+8

−

uεs | (t

ti+1 −2ε

2 |uεt

−

uεs | (t

H− 32

− s)

dt

ds.

298

5. Fractional Brownian motion

The ﬁrst and second integrals converge to zero, due to the estimate c |uεt − uεs | ≤ |t − s|. ε On the other hand, the third and fourth term of the above expression converge to zero because uεt is bounded. Therefore we have proved that 2

E u − uε K −→ 0

as

ε → 0.

Finally, it is easy to see by the same arguments that we also have

T

2

Dr u − Dr uε K dr −→ 0

E 0

ε → 0.

as

Thus the proof is complete. Now we are ready to prove Proposition 5.2.4.

Proof of Proposition 5.2.4: From the properties of the divergence operator, and applying Fubini’s theorem we have (2ε)−1

T

us (Bs+ε − Bs−ε ) ds = (2ε)−1

0

+(2ε)−1 =

(2ε)−1

T

#

Dus , 1[s−ε,s+ε] H 0 r+ε T

r −ε

0

+(2ε)−1

$

0

T

#

0

T

δ us 1[s−ε,s+ε] (·) ds

ds

us ds dBr

$ D· us , 1[s−ε,s+ε] (·) H ds

T

uεr dBr + B ε .

= 0

By our hypothesis we get that B ε converges to T rDu in probability as T ε ↓ 0. In order to see that 0 uεr dBr converges to δ (u) in L2 (Ω) as ε tends to zero, we will show that uε converges to u in the norm of D1,2 (HK ). Fix δ > 0. We have already noted that the deﬁnition of the space D1,2 (HK ) implies that there is a bounded simple HK -valued processes ϕ as in (5.48) such that T 2 2 ||Dr u − Dr ϕ||K dr ≤ δ. (5.52) E u − ϕK + E 0

5.2 Stochastic calculus with respect to fBm

299

Therefore, Lemma 5.2.1 implies that for ε small enough, E u −

2 uε K

T

2

||Dr (u − uε )||K dr

+E 0

2

≤ cE u − ϕK + cE

T

0

2

||Dr (u − ϕ)||K dr

T

2

+cE ϕ − ϕε K + cE

0

2

+cE ϕε − uε K + cE ≤ 2cδ + cE ϕ − ε

2 u ε K

2

||Dr (ϕ − ϕε )||K dr T

0

2

||Dr (ϕε − uε )||K dr

T

+ cE 0

2

||Dr (ϕε − uε )||K dr.

(5.53)

We have

T

2

E (ϕεs − uεs ) (T − s)2H−1 ds 0

T

≤ ≤ 0

T

2

s+ε

(ϕr − ur ) dr (T − s)2H−1 ds (r+ε)∧T 1 2 2H−1 E (ϕr − ur ) (T − s) ds dr. 2ε (r−ε)∨0 E

0

1 2ε

s−ε

From property (i) it follows that

(2ε)−1

(r+ε)∧T

(r −ε)∨0

" ! K(T, t)2 dt ≤ c (T − r)−2α + r−2α .

Hence, by the dominated convergence theorem and condition (4.21) we obtain ε↓0

≤

T

T

2

E (ϕεs − uεs ) K(T, s)2 ds

lim sup 0

2

E (ϕs − us ) K(T, s)2 ds ≤ δ. 0

(5.54)

300

5. Fractional Brownian motion

On the other hand,

T

2

T

|ϕεt

E 0

1 ≤ E 2ε 1 = E 2ε ≤E

T

T

T

T +ε

−ε T +ε

−ε

T

+

uεs | (t

s+ε

(ϕ − u)

t−θ

s

s+ε

T +r−s

T +ε

r (r+ε)∧T

dtt

ds

2 H− 32 − (ϕ − u)s−θ (t − s) dtdθθ ds 2 H− 32

|(ϕ − u)t − (ϕ − u)r | (t − r)

dtdrr

ds

2 H− 32

|(ϕ − u)t − (ϕ − u)r | (t − r)

(r −ε) ε ∨0 T +ε

− s)

H− 32

r

s−ε

0

s−ε

0

ε

−ε

0

1 = 2E 4ε

−

ϕεs

s

1 ≤ 2E 4ε

−

uεt

T +ε

dtt

drds 2

|ϕt − ut − ϕr + ur | (t − r)

r

H− 32

dtt

dsdr

2 H− 32

|ϕt − ut − ϕr + ur | (t − r)

dtt

dr.

(5.55)

r

By (5.54) and (5.55) we obtain 2

lim sup E ϕε − uε K ≤ 2δ. ε↓0

By a similar argument,

T

lim sup E ε↓0

0

2

||Dr (ϕε − uε )||K dr ≤ 2δ.

Since δ is arbitrary, uε converges to u in the norm of D1,2 (HK ) as ε ↓ 0, T and, as a consequence, 0 uεr dBr converges in L2 (Ω) to δ (u) . Thus the proof is complete. Consider the particular case of the process ut = F (Bt ), where F is a continuously diﬀerentiable function satisfying the growth condition (5.42). If H > 14 , the process F (Bt ) the process belongs to D1,2 (HK ). Moreover, TrDu exists and T

TrDu = H

F (Bt )t2H−1 dt.

0

As a consequence we obtain

T

F (Bt )0 dBt = 0

T

F (Bt )dBt + H 0

0

T

F (Bt )t2H−1 dt.

5.2 Stochastic calculus with respect to fBm

(C) Ito’s ˆ formulas for the divergence integral in the case H <

301

1 2

An Itˆ o’s formula similar to (5.44) was proved in [9] for general Gaussian t processes of Volterra-type of the form Bt = 0 K(t, s)dW Ws , where K(t, s) is a singular kernel. In particular, the process Bt can be a fBm with Hurst oˆ’s formula for the inparameter 14 < H < 12 . Moreover, in this paper, an Itˆ t deﬁnite divergence process Xt = 0 us dBs similar to (5.45) was also proved. On the other hand, in the case of the fractional Brownian motion with ˆ formula for the indeﬁnite symmetric Hurst parameter 14 < H < 12 , an Ito’s t integral Xt = 0 us dBs has been proved in [7] assuming again 14 < H < 12 . Let us explain the reason for the restriction 14 < H. In order to deﬁne the T divergence integral 0 F (Bs )dBs , we need the process F (Bs ) to belong to L2 (Ω; H). This is clearly true, provided F satisﬁes the growth condition (5.42), because F (Bs ) is H¨¨older continuous of order H − ε > 12 − H if ε < 2H − 12 . If H ≤ 14 , one can show (see [66]) that P (B ∈ H) = 0, and the space D1,2 (H) is too small to contains processes of the form F (Bt ). Following the approach of [66] we are going to extend the domain of the divergence operator to processes whose trajectories are not necessarily in the space H. Using (5.31) and applying the integration by parts formula for the fractional calculus (A.17) we obtain for any f, g ∈ H f, gH

∗ ∗ = KH f, KH gL2 ([0,T ]) @ 1 A 1 1 1 1 1 −H −H = d2H s 2 −H DT2 − sH− 2 f, s 2 −H DT2 − sH− 2 g 2 L ([0,T ]) @ A 1 1 1 2 H− 12 12 −H 2 −H 1−2H 2 −H H− 2 = dH f, s s D0+ s DT − s g . L2 ([0,T ])

∗ This implies that the adjoint of the operator KH in L2 ([0, T ]) is 1 1 ∗,a 1 1 −H 2 −H 1−2H KH f (s) = dH s 2 −H D0+ s DT2 − sH− 2 f. ∗,a −1 2 ∗ −1 KH ) (L ([0, T ])). Denote by SH the space of Set H2 = (KH smooth and cylindrical random variables of the form

F = f (B(φ1 ), . . . , B(φn )),

(5.56)

where n ≥ 1, f ∈ Cb∞ (Rn ) (f and all its partial derivatives are bounded), and φi ∈ H2 . Deﬁnition 5.2.3 Let u = {ut , t ∈ [0, T ]} be a measurable process such that T 2 ut dt < ∞. E 0

302

5. Fractional Brownian motion

We say that u ∈ Dom∗ δ if there exists a random variable δ(u) ∈ L2 (Ω) such that for all F ∈ SH we have ∗,a ∗ E(ut KH KH Dt F )dt = E(δ(u)F ). R

This extended domain of the divergence operator satisﬁes the following elementary properties: 1. Domδ ⊂ Dom∗ δ, and δ restricted to Domδ coincides with the divergence operator. 2. If u ∈ Dom∗ δ then E(u) belongs to H. 3. If u is a deterministic process, then u ∈ Dom∗ δ if and only if u ∈ H. This extended domain of the divergence operator leads to the following version of Ito’s ˆ formula for the divergence process, established by Cheridito and Nualart in [66]. Theorem 5.2.2 Suppose that F is a function of class C 2 (R) satisfying the growth condition (5.42). Then for all t ∈ [0, T ], the process {F (Bs )1[0,t] (s)} belongs to Dom∗ δ and we have F (Bt ) = F (0) +

t

F (Bs )dBs + H

0

Proof:

t

F (Bs )s2H−1 ds.

(5.57)

0

Notice that F (Bs )1[0,t] (s) ∈ L2 ([0, T ] × Ω) and F (Bt ) − F (0) − H

t

F (Bs )s2H−1 ds ∈ L2 (Ω) .

0

Hence, it suﬃces to show that for any F ∈ SH E(F (Bs )1[0,t] (s), Ds F H t 2H−1 =E F (Bt ) − F (0) − H F (Bs )s ds F .

(5.58)

0

Take F = Hn (B(ϕ)), where ϕ ∈ H2 and Hn is the nth-Hermite polynomial. We have Dt F = Hn−1 (B(ϕ))ϕt Hence, (5.58) can be written as E Hn−1 (B(ϕ))F (Bs )1[0,t] (s), ϕs H t = E((F (Bt ) − F (0) − H F (Bs )s2H−1 ds)H Hn (B(ϕ))) 0

5.2 Stochastic calculus with respect to fBm

303

Using (5.33) we obtain

t

1

1

−H

−H

E (F (Bs )H Hn−1 (B(ϕ))) (D+2 D−2 ϕ)(s)ds 0 t F (Bs )s2H−1 ds)H Hn (B(ϕ))). (5.59) = E((F (BtH ) − F (0) − H e2H

0

In order to show (5.59) we will replace F by Fk (x) = k

1

−1

F (x − y)ε(ky)dy,

where ε is a nonnegative smooth function supported by [−1, 1] such that 1 ε(y)dy = 1. −1 We will make use of the following equalities: Hn (B(ϕ))) E(F (Bt )H

1 E(F (Bt )δ n (ϕ⊗n )) n! 1 EDn (F (Bt )), ϕ⊗n H⊗n n! 1 E(F (n) (Bt ))1(0,t] , ϕnH . n!

= = =

2 1 y . Note that Let p(σ, y) := (2πσ)− 2 exp − 2σ and s ∈ (0, t], d E(F (n) (Bs )) ds

=

d ds

∂p ∂σ

=

2 1∂ p 2 ∂y 2 .

For all n ≥ 0

p(s2H , y)F (n) (y)dy R

∂p 2H (s , y)2Hs2H−1 F (n) (y)dy = R ∂σ 2 ∂ p 2H (s , y)F (n) (y)dy = Hs2H−1 2 R ∂y 2H−1 p(s2H , y)F (n+2) (y)dy = Hs R

= Hs2H−1 E(F (n+2) (Bs )).

(5.60)

For n = 0 the left hand side of (5.59) is zero. On the other hand it follows from (5.60) that E (F (Bt )) − F (0) − H

t

E(F (Bs ))s2H−1 ds = 0

0

This shows that (5.59) is valid for n = 0.

304

5. Fractional Brownian motion

Fix n ≥ 1. Equation (5.60) implies that for all s ∈ (0, t], d E(F (n) (Bs ))1(0,s] , ϕnH ds

= Hs2H−1 E(F (n+2) (Bs ))1(0,s] , ϕnH +e2H nE(F (n) (Bs )) 1

−H

2 ×1(0,s] , ϕn−1 H (D+

1

−H

D−2

ϕ)(s).

It follows that E(F

(n)

t

E(F (n+2) (Bs ))1(0,s] , ϕnH s2H−1 ds t E(F (n) (Bs )) +e2H n

(Bt ))1(0,s] , ϕnH

= H

0

0 1

−H

2 ×1(0,s] , ϕn−1 H (D+

1

−H

D−2

ϕ)(s), (5.61)

(5.61) is equivalent to (5.59) because E(F (n) (Bt ))1(0,s] , ϕnH = n!E(F (Bt )H Hn (B(ϕ))), E(F (n) (Bs ))1(0,s] , ϕn−1 = (n − 1) !E(F (Bs )H(n−1) (B(ϕ))), H and E(F (n+2) (Bs ))1(0,s] , ϕnH = n!E(F (Bs )H Hn (B(ϕ))). This completes the proof (5.59) for the function Fk . Finally it suﬃces to let k tend to inﬁnity. (D) Local time and Tanaka’s formula for fBm Berman proved in [22] that that fractional Brownian motion B = {Bt , t ≥ 0} has a local time lta continuous in (a, t) ∈ R × [0, ∞) which satisﬁes the occupation formula t g(Bs )ds = g(a)lta da (5.62) R

0

for every continuous and bounded function g on R. Moreover, lta is increasing in the time variable. Set Lat

t

s2H−1 la (ds).

= 2H 0

It follows from (5.62) that

t

g(Bs )s2H−1 ds =

2H 0

R

g(a)Lat da.

5.2 Stochastic calculus with respect to fBm

305

This means that a → Lat is the density of the occupation measure

t

1C (Bs )s2H−1 ds,

µ(C) = 2H 0

where C is a Borel subset of R. Furthermore, the continuity property of lta implies that Lat is continuous in (a, t) ∈ R × [0, ∞). As an extension of the Itˆo’s formula (5.57), the following result has been proved in [66]: Theorem 5.2.3 Let 0 < t < ∞ and a ∈ R. Then 1{Bs >a} 1[0,t] (s) ∈ Dom∗ δ , and

t

(Bt − a)+ = (−a)+ + 0

1 1{Bs >a} dBs + Lat . 2

(5.63)

This result can be considered as a version of Tanaka’s formula for the fBm. In [69] it is proved that for H > 13 , the process 1{Bs >a} 1[0,t] (s) belongs to Domδ and (5.63) holds. ¨ continuous paths of order δ < 1 − H in The local time lta has Holder in the space variable, provided H ≥ 13 (see time, and of order γ < 1−H 2H a Table 2 in [117]). Moreover, lt is absolutely continuous in a if H < 13 , it is continuously diﬀerentiable if H < 15 , and its smoothness in the space variable increases when H decreases. In a recent paper, Eddahbi, Lacayo, Sol´e, Tudor and Vives [88] have proved that lta ∈ Dα,2 for all α < 1−H 2H . That means, the regularity of the local time lta in the sense of Malliavin calculus is the same order as its H¨ older continuity in the space variable. This result follows from the Wiener chaos expansion (see [69]): lta =

∞ n=0

t

0

s−nH p(s2H , a)H Hn (as−H )IIn 1[o,s]

⊗n

ds.

In fact, the series ∞

=

n=0 ∞ n=0

(

t

α

(1 + n) E

−nH

s 0

t (1 + n)α n! 0

×RH (r, s)n drds

0

t

2H

p(s

−H

, a)H Hn (as

)IIn 1[o,s]

⊗n

2 ) ds

(sr)−nH p(s2H , a)p(r2H , a)H Hn (as−H )H Hn (ar−H )

306

5. Fractional Brownian motion

is equivalent to ∞ n=1 ∞

=

− 12 +α

t

t

n

0

1

n− 2 +α

RH (u, v)(uv)−nH−1 dudv

0 1

RH (1, z)z −nH−1 dz.

0

n=0

Then, the result follows from the estimate 1 1 −nH−1 RH (1, z)z dz ≤ Cn− 2H . 0

Exercises 5.2.1 Show the tranfer principle stated in Propositions 5.2.1 and 5.2.2. 5.2.2 Show the inequality (5.34). 5.2.3 Show the estimate (5.47). 5.2.4 Deduce the Wiener chaos expansion of the local time lta .

5.3 Stochastic diﬀerential equations driven by a fBm In this section we will establish the existence and uniqueness of a solution for stochastic diﬀerential equations driven by a fractional Brownian motion with Hurst parameter H > 12 , following an approach based on the fractional calculus. We ﬁrst introduce a notion of Stieltjes integral based on the fractional integration by parts formula (A.17).

5.3.1 Generalized Stieltjes integrals Given a function g : [0, T ] → R, set gT − (s) = g(s) − limε↓0 (T − ε) provided this limit exists. Take p, q ≥ 1 such that p1 + 1q ≤ 1 and 0 < α < 1. Suppose that f and g are functions on [0, T ] such that g(T −) exists, α q (Lp ) and gT − ∈ IT1−α f ∈ I0+ − (L ). Then the generalized Stieltjes integral of f with respect to g is deﬁned by (see [357])

T

fs dgs = 0

0

T α D0+ fa+ (s) DT1−α − gT − (s) ds.

(5.64)

In [357] it is proved that this integral coincides with the Riemann-Stieltjes integral if f and g are H¨¨older continuous of orders α and β with α + β > 1.

5.3 Stochastic diﬀerential equations driven by a fBm

307

Fix 0 < α < 12 . Denote by W0α,∞ (0, T ) the space of measurable functions f : [0, T ] → R such that t |f (t) − f (s)| f α,∞ := sup |f (t)| + < ∞. α+1 ds (t − s) t∈[0,T ] 0 We have, for all 0 < ε < α C α+ε (0, T ) ⊂ W0α,∞ (0, T ) ⊂ C α−ε (0, T ). Denote by WT1−α,∞ (0, T ) the space of measurable functions g : [0, T ] → R such that t |g(t) − g(s)| |g(y) − g(s)| + dy < ∞. g1−α,∞,T := sup (t − s)1−α (y − s)2−α 0<s
Λα (g) :=

Finally, denote by W0α,1 (0, T ) the space of measurable functions f on [0, T ] such that T T s |f (s)| |f (s) − f (y)| f α,1 := ds + dy ds < ∞. α s (s − y)α+1 0 0 0 Lemma 5.3.1 Fix 0 < α < 12 and consider two functions g ∈ WT1−α,∞ (0, T ) and f ∈ W0α,1 (0, T ). Then the generalized Stieltjes integral

t

T

fs dgs := 0

0

fs 1[0,t] (s)dgs

exists for all t ∈ [0, T ] and for any s < t we have

t

fu dgu − 0

s

fu dgu = 0

t

1−α α Ds+ (f ) (r) Dt− gt− (r) dr.

s

This lemma follows easily from the deﬁnition of the fractional derivative. We have

308

5. Fractional Brownian motion

t fs dgs ≤ Λα (g) f α,1 . 0

Indeed, t t α 1−α D0+ f (s) Dt− gt− (s) ds fs dgs = 0 0 1−α t α D0+ f (s) ds ≤ sup Dt− gt− (s) 0≤s≤t≤T

0

≤ Λα (g) f α,1 . The following proposition is the main estimate for the generalized Stieltjes integral. Proposition 5.3.1 Fix 0 < α < 12 . Given two functions g ∈ WT1−α,∞ (0, T ) and f ∈ W0α,1 (0, T ) we set t ht = fs dgs . 0

Then for all s < t ≤ T we have t |ht − hs | |ht | + ds ≤ α+1 0 (t − s)

t (1) (t − r)−2α + r−α Λα (g) cα,T 0 r |ffr − fy | × |ffr | + dy dr, (5.65) α+1 0 (r − y)

(1)

where cα,T is a constant depending on α and T . Proof: Using the deﬁnition and additivity property of the indeﬁnite integral we obtain t t 1−α α f dg = Ds+ (f ) (r) Dt− gt− (r) dr |ht − hs | = s s t |ffr | ≤ Λα (g) dr (r − s)α s t r |ffr − fy | +α dydr . (5.66) α+1 s s (r − y) Taking s = 0 we obtain the desired estimate for |ht |. Multiplying (5.66) by (t − s)−α−1 and integrating in s yields t t |ht − hs | ds ≤ Λα (g) (t − s)−α−1 (5.67) α+1 0 (t − s) 0 t r t |ffr | |ffr − fy | dr + α dydr ds. × α α+1 s (r − s) s s (r − y)

5.3 Stochastic diﬀerential equations driven by a fBm

By the substitution s = r − (t − r)y we have r ∞ (t − s)−α−1 (r − s)−α ds ≤ (t − r)−2α (1 + y)−α−1 y −α dy 0

309

(5.68)

0

and, on the other hand, y ! " (t − s)−α−1 ds = α−1 (t − y)−α − t−α ≤ α−1 (t − y)−α .

(5.69)

0

Substituting (5.68) and (5.69) into (5.67) yields & t t |ht − hs | |ffr | (1) ds ≤ Λα (g) cα dr α+1 (t − s) (t − r)2α 0 0 ' t r |f (r) − f (y)| −α + (t − y) dydr , (r − y)α+1 0 0

where c(1) α =

∞

(1 + y)−α−1 y −α dy = B(2α, 1 − α).

0

WT1−α,∞ (0, T )

As a consequence of this estimate, if f ∈ we have t (2) fs dgs ≤ Λα (g) cα,T (t − s)1−α f α,∞ , (5.70) s

and

0

·

fs dgs

(3)

≤ Λα (g) cα,T f α,∞ .

(5.71)

α,∞

5.3.2 Deterministic diﬀerential equations Let 0 < α < 12 be ﬁxed. Let g ∈ WT1−d,∞ (0, T ; Rd ). Consider the deterministic diﬀerential equation on Rd t d xt = x0 + b(s, xs )ds + σ j (s, xs ) dgsj , t ∈ [0, T ] , (5.72) 0

j=1

where x0 ∈ R . Let us introduce the following assumptions on the coeﬃcients: m

H1 σ(t, x) is diﬀerentiable in x, and there exist some constants 0 < β, δ ≤ 1 and for every N ≥ 0 there exists MN > 0 such that the following properties hold: |σ(t, x) − σ(t, y)| ≤ M0 |x − y|,

∀ x ∈ Rm , ∀ t ∈ [0, T ] ,

|∂ ∂xi σ(t, x) − ∂xi σ(t, y)| ≤ MN |x − y|δ , ∀ |x| , |y| ≤ N, ∀t ∈ [0, T ] , |σ(t, x) − σ(s, x)| + |∂ ∂xi σ(t, x) − ∂xi σ(s, x)| ≤ M0 |t − s|β , ∀ x ∈ Rm , ∀ t, s ∈ [0, T ] .

310

5. Fractional Brownian motion

for each i = 1, . . . , m. H2 The coeﬃcient b(t, x) satisﬁes for every N ≥ 0 |b(t, x) − b(t, y)|

≤ LN |x − y|, ∀ |x| , |y| ≤ N, ∀t ∈ [0, T ] ,

|b(t, x)|

≤

L0 |x| + b0 (t) , ∀x ∈ Rm , ∀t ∈ [0, T ] ,

where b0 ∈ Lρ (0, T ; Rm ), with ρ ≥ 2 and for some constant LN > 0. Theorem 5.3.1 Suppose that the coeﬃcients σ and b satisfy the assumptions H1 and H2 with ρ = α1 , 0 < β, δ ≤ 1 and 0 < α < α0 = δ . Then Equation (5.72) has a unique continuous solution min 12 , β, δ+1 such that xi ∈ W0α,∞ (0, T ) for all i = 1, . . . , m. Sketch of the proof: f Suppose d = m = 1. Fix λ > 1 and deﬁne the seminorm in W0α,∞ (0, T ) by f α,λ = sup

e−λt |fft | +

t∈[0,T ]

t

0

|fft − fs | ds . (t − s)α+1

Consider the operator L deﬁned by t t (Lf )t = x0 + b(s, fs )ds + σ (s, fs ) dgs . 0

0

There exists λ0 such that for λ ≥ λ0 we have Lf α,λ ≤ |x0 | + 1 +

1 f α,λ . 2

Hence, the operator L leaves invariant the ball B0 of radius 2 (|x0 | + 1) in the norm ·α,λ0 of the space W0α,∞ (0, T ). Moreover, there exists a constant C depending on g such that for any λ ≥ 1 and u, v ∈ B0 L (u) − L (v)α,λ ≤

C λ

1−2α

(1 + ∆ (u) + ∆ (v)) u − vα,λ ,

where

∆ (u) = sup r∈[0,T ]

0

r

|ur − us |δ ds. (r − s)α+1

A basic ingredient in the proof of this inequality is the estimate |σ (r, fr ) − σ (s, fs ) − σ (r, hr ) + σ (s, hs )| ≤ M0 |ffr − fs − hr + hs | + M0 |ffr − hr |(r − s)β δ δ + MN |ffr − hr | |ffr − fs | + |hr − hs | ,

(5.73)

5.3 Stochastic diﬀerential equations driven by a fBm

311

which is an immediate consequence of the properties of the function σ. The seminorm ∆ is bounded on L (B0 ), and, as a consequence, (5.74) implies that L is a contraction operator in L (B0 ) with respect to a diﬀerent norm ·α,λ2 for a suitable value of λ2 > 1. Finally, the the existence of a solution follows from a suitable ﬁxed point argument (see Lemma 5.3.2). The uniqueness is proved again using the main estimate (5.65). The following lemma has been used in the proof of Theorem 4.1.1 (for its proof see [254]). Lemma 5.3.2 Let (X, ρ) be a complete metric space and ρ0 , ρ1 , ρ2 some metrics on X equivalent to ρ. If L : X → X satisﬁes: i) there exists r0 > 0, x0 ∈ X such that if B0 = {x ∈ X : ρ0 (x0 , x) ≤ r0 } then L (B0 ) ⊂ B0 , ii) there exists ϕ : (X, ρ) → [0, +∞] lower semicontinuous function and some positive constants C0 , K0 such that denoting Nϕ (a) = {x ∈ X : ϕ(x) ≤ a} a) b) iii)

C0 ) , L (B0 ) ⊂ Nϕ (C C0 ) ∩ B0 , ρ1 (L (x) , L (y)) ≤ K0 ρ1 (x, y) , ∀x, y ∈ Nϕ (C

there exists a ∈ (0, 1) such that ρ2 (L (x) , L (y)) ≤ a ρ2 (x, y) , ∀x, y ∈ L (B0 ) ,

then there exists x∗ ∈ L (B0 ) ⊂ X such that x∗ = L (x∗ ) . Estimates of the solution Suppose that the coeﬃcient σ satisﬁes the assumptions of the Theorem 4.1.1 and γ |σ (t, x)| ≤ K0 (1 + |x| ) , (5.74) where 0 ≤ γ ≤ 1. Then, the solution f of Equation (5.72) satisﬁes f α,∞ ≤ C1 exp (C C2 Λα (g)κ ) , where

⎧ ⎨ κ=

⎩

1 1−2α γ > 1−2α 1 1−α

(5.75)

if γ=1 if 1−2α 1−α ≤ γ < 1 if 0 ≤ γ < 1−2α 1−α

and the constants C1 and C2 depend on T , α, and the constants that appear in conditions H1, H2 and (5.74). The proof of (5.75) is based on the following version of Gronwall lemma:

312

5. Fractional Brownian motion

Lemma 5.3.3 Fix 0 ≤ α < 1, a, b ≥ 0. Let x : [0, ∞) → [0, ∞) be a continuous function such that for each t t (t − s)−α s−α xs ds. (5.76) xt ≤ a + btα 0

Then

∞

Γ(1 − α)n+1 tn(1−α) . Γ[(n + 1)(1 − α)] n=1 ≤ adα exp cα tb1/(1−α) ,

xt ≤ a + a

bn

(5.77)

where ca and dα are positive constants depending only on α (as an example, 1/(1−α) one can set cα = 2 (Γ(1 − α)) and dα = 4e2 Γ(1−α) 1−α ). This implies that there exists a constants cα , dα > 0 such that xt ≤ adα exp cα tb1/(1−α) .

5.3.3 Stochastic diﬀerential equations with respect to fBm Let B = {Bt , t ≥ 0} be a d-dimensional fractional Brownian motion of Hurst parameter H ∈ 12 , 1 . This means that the components of B are independent fBm with the same Hurst parameter H. Consider the equation on Rm t d t j Xt = X0 + σ j (s, Xs ) ◦ dBs + b(s, Xs )ds, t ∈ [0, T ] , (5.78) j=1

0

0

where X0 is an m-dimensional random variable. The integral with respect to B is a path-wise Riemann-Stieltjes integral, and we know that this ino¨lder continuous tegral exists provided that the process σ j (s, Xs ) has H¨ trajectories of order larger that 1 − H. Choose α such that 1 − H < α < 12 . By Fernique’s theorem, for any 0 < δ < 2 we have E exp Λα (B)δ < ∞. As a consequence, if u = {ut , t ∈ [0, T ]} is a stochastic process whose trajectories belong to the space WTα,1 (0, T ), almost surely, the path-wise T generalized Stieltjes integral integral 0 us ◦ dBs exists and we have the estimate T us ◦ dBs ≤ G uα,1 . 0 Moreover, if the trajectories of the process u belong to the space W0α,∞ (0, T ), t then the indeﬁnite integral Ut = 0 us ◦ dBs is H¨o¨lder continuous of order 1 − α, and its trajectories also belong to the space W0α,∞ (0, T ).

5.4 Vortex ﬁlaments based on fBm

313

Consider the stochastic diﬀerential equation (5.78) on Rm where the process B is a d-dimensional fBm with Hurst parameter H ∈ 12 , 1 and X0 is an m-dimensional random variable. Suppose that the coeﬃcients σ ij , bi : Ω × [0, T ] × Rm → R are measurable functions satisfying conditions H1 and H2, where the constants MN and LN may depend on ω ∈ Ω, and β > 1 − H, δ > H1 − 1. Fix α such that 1 δ , β, 1 − H < α < α0 = min 2 δ+1 and α ≤ ρ1 . Then the stochastic equation (5.65) has a unique continuous solution such that X i ∈ W0α,∞ (0, T ) for all i = 1, . . . , m. Moreover the solution is H¨ o¨lder continuous of order 1 − α. Assume that X0 is bounded and the constants do not depend on ω. Suppose that γ |σ (t, x)| ≤ K0 (1 + |x| ) , where 0 ≤ γ ≤ 1. Then, Xα,∞ ≤ C1 exp (C C2 Λα (B)κ ) . Hence,for all p ≥ 1 p E Xα,∞ ≤ C1p E (exp (pC C2 Λα (B)κ )) < ∞ provided κ < 2, that is, γ 1 + ≤H 4 2 and 1−H <α< • If γ = 1 this means α < • If γ < 2 −

1 H

1 4

1 γ − . 2 4

and H > 34 .

we can take any α such that 1 − H < α < 12 .

Exercises 5.3.1 Show the estimates (5.70) and (5.71). 5.3.2 Show Lemma 5.3.1.

5.4 Vortex ﬁlaments based on fBm The observations of three-dimensional turbulent ﬂuids indicate that the vorticity ﬁeld of the ﬂuid is concentrated along thin structures called vortex ﬁlaments. In his book Chorin [67] suggests probabilistic descriptions of

314

5. Fractional Brownian motion

vortex ﬁlaments by trajectories of self-avoiding walks on a lattice. Flandoli [102] introduced a model of vortex ﬁlaments based on a three-dimensional Brownian motion. A basic problem in these models is the computation of the kynetic energy of a given conﬁguration. Denote by u(x) the velocity ﬁeld of the ﬂuid at point x ∈ R3 , and let ξ = curlu be the associated vorticity ﬁeld. The kynetic energy of the ﬁeld will be ξ(x) · ξ(y) 1 1 2 dxdy. (5.79) |u(x)| dx = H= 2 R3 8π R3 R3 |x − y| We will assume that the vorticity ﬁeld is concentrated along a thin tube centered in a curve γ = {γ t , 0 ≤ t ≤ T }. Moreover, we will choose a random model and consider this curve as the trajectory of a three-dimensional fractional Brownian motion B = {Bt , 0 ≤ t ≤ T }. This can be formally expressed as T

R3

·

δ(x − y − Bs )B s ds ρ(dy),

ξ(x) = Γ

(5.80)

0

where Γ is a parameter called the circuitation, and ρ is a probability measure on R3 with compact support. Substituting (5.80) into (5.79) we derive the following formal expression for the kynetic energy: Hxy ρ(dx)ρ(dy), (5.81) H= R3

R3

where the so-called interaction energy Hxy is given by the double integral Hxy =

3 1 Γ2 T T ◦ dBsi ◦ dBti . 8π i=1 0 0 |x + Bt − y − Bs |

(5.82)

We are interested in the following problems: Is H a well deﬁned random variable? Does it have moments of all orders and even exponential moments? In order to give a rigorous meaning to the double integral (5.82) let us −1 introduce the regularization of the function |·| : −1

σ n = |·|

∗ p1/n ,

(5.83)

where p1/n is the Gaussian kernel with variance n1 . Then, the smoothed interaction energy 3 T Γ2 T n i Hxy = σ n (x + Bt − y − Bs ) ◦ dBs ◦ dBti , (5.84) 8π i=1 0 0

5.4 Vortex ﬁlaments based on fBm

315

is well deﬁned, where the integrals are path-wise Riemann-Stieltjes integrals. Set Hn = Hnxy ρ(dx)ρ(dy). (5.85) R3

R3

The following result has been proved in [255]: Theorem 5.4.1 Suppose that the measure ρ satisﬁes 1 |x − y|1− H ρ(dx)ρ(dy) < ∞. R3

(5.86)

R3

Let Hnxy be the smoothed interaction energy deﬁned by (5.84). Then Hn deﬁned in (5.85) converges, for all k ≥ 1, in Lk (Ω) to a random variable H ≥ 0 that we call the energy associated with the vorticity ﬁeld (5.80). Brownian motion. In If H = 12 , fBm B is a classical three-dimensional this case condition (5.86) would be R3 R3 |x − y|−1 ρ(dx)ρ(dy) < ∞, which is the assumption made by Flandoli [102] and Flandoli and Gubinelli [103]. In this last paper, using Fourier approach and Itˆ oˆ’s stochastic calculus, the authors show that Ee−βH < ∞ for suﬃciently small negative β. The proof of Theorem 5.4.1 is based on the stochastic calculus of variations with respect to fBm and the application of Fourier transform. Sketch of the proof: f The proof will be done in two steps: Step 1 (Fourier transform) Using 1 = |z| we get

(2π)3 R3

σ n (x) =

R3

e−iξ,z dξ |ξ|2

|ξ|−2 eiξ,x−|ξ|

2

/2n

dξ.

Substituting this expression in (5.84), we obtain the following formula for the smoothed interaction energy 3 T T 2 −|ξ|2 /2n e Γ eiξ,x+Bt −y−Bs ◦ dBsj ◦ dBtj Hnxy = 2 8π j=1 0 0 |ξ| 3 R 2 2 Γ 2 = |ξ|−2 eiξ,x−y−|ξ| /2n Y Yξ C dξ, (5.87) 8π R3 where

Yξ = 2

and Y Yξ C =

3 i=1

T

eiξ,Bt ◦ dBt

0 i

Yξi Yξ . Integrating with respect to ρ yields

316

5. Fractional Brownian motion

Γ2 H = 8π

Y Yξ C |ξ|−2 | ρ(ξ)| e−|ξ| 2

n

R3

2

2

/2n

dξ ≥ 0.

(5.88)

From Fourier analysis and condition (5.86) we know that

R3

R3

1

|x − y|1− H ρ(dx)ρ(dy) = CH

1

| ρ(ξ)| |ξ| H −4 dξ < ∞. 2

R3

(5.89)

Then, taking into account (5.89) and (5.88), in order to show the convergence in Lk (Ω) of Hn to a random variable H ≥ 0 it suﬃces to check that 1 2k (5.90) E Y Yξ C ≤ Ck 1 ∧ |ξ|k( H −2) . Step 2 (Stochastic calculus) We will present the main arguments for the proof of the estimate (5.90) for k = 1. Relation (5.37) applied to the process T ut = eiξ,Bt allows us to decompose the path-wise integral Yξ = 0 eiξ,Bt ◦ dBt into the sum of a divergence plus a trace term:

T

Yξ =

iξ,Bt

e

T

dBt + H

0

iξeiξ,Bt t2H−1 dt.

(5.91)

0

On the other hand, applying the three dimensional version of Itˆ oˆ’s formula (5.44) we obtain 3

eiξ,BT = 1 +

T

iξ j eiξ,Bt δBtj − H

0

j=1

T

t2H−1 |ξ|2 eiξ,Bt dt. (5.92)

0

Multiplying both members of (5.92) by iξ|ξ|−2 and adding the result to (5.91) yields

T

iξ,Bt

Yξ = pξ

e

−

dBt

0

where pξ (v) = v −

ξ |ξ|2

iξ iξ,BT (1) (2) e − 1 := Yξ + Yξ , |ξ|2 ⊥

ξ, v is the orthogonal projection of v on ξ . It (1)

suﬃces to derive the estimate (5.90) for the term Yξ . Using the duality relationship (1.42) for each j = 1, 2, 3 we can write E

(1),j (1),j Yξ Yξ

6

=E

iξ,B·

e

, pjξ D·

pjξ

T

7 −iξ,Bt

e 0

dBt

. H

(5.93)

5.4 Vortex ﬁlaments based on fBm

317

The commutation relation D(δ(u)), hH = u, hH + δ(Du, hH ) implies T T k j k −iξ,Bt D e dB = e−iξ,Br δ + (−iξ k ) 1 (r)e−iξ,Bt dB j . k,j

t

r

0

[0,t]

0

Applying the projection operators yields T

pjξ Dr

pjξ

−iξ,Bt

e

t

ξ∗ξ = e I− 2 |ξ| j,j j 2 ξ = e−iξ,Br 1 − . |ξ|2

dBt

0

−iξ,Br

Notice that the term involving derivatives in the expectation (5.93) vanishes. This cancellation is similar to what happens in the computation of the variance of the divergence of an adapted process, in the case of the Brownian motion. Hence, 3

(1),j (1),j E Yξ Yξ

@ =

2E

=

2αH

j=1

e−iξ,B· , e−iξ,B· T

T

0

=

T

H

E eiξ,Bs −Br |s − r|2H−2 dsdr

0 T

2αH 0

A

e−

|s−r|2H 2

|ξ|2

|s − r|2H−2 dsdr,

0

1 H

which behaves as |ξ| −2 as |ξ| tends to inﬁnity. This completes the proof of the desired estimate for k = 1. In the general case k ≥ 2 the proof makes use of the local nondeterminism property of fBm: 2H (Bti − Bsi ) ≥ kH (ti − si ) . Var i

i

Decomposition of the interaction energy Assume

1 2

< H < 23 . For any x = y, set

F H xy =

3 i=1

0

T

0

t

1 ◦ dBsi |x + Bt − y − Bs |

◦ dBti .

(5.94)

2 n F F Then H xy exists as the limit in L (Ω) of the sequence Hxy deﬁned using −1 the approximation σ n (x) of |x| introduced in (5.83) and the following

318

5. Fractional Brownian motion

decomposition holds 3 T t 1 F dBri dBti . = H xy |x − y + B − B | t r 0 i=1 0 T t −H 2 δ 0 (x − y + Bt − Br )(t − r)2(2H−1) drdt. 0 0 t T 1 2H−2 (t − r) dr dt +H(2H − 1) 0 0 |x − y + Bt − Br | T 1 1 (T − r)2H−2 + r2H−11 dr. +H |x − y + BT − Br | |x − y + Br | 0 F Notice that in comparison with Hxy , in the deﬁnition of H xy we chose to deal with the half integral over the domain {0 ≤ s ≤ t ≤ T } , 2

Γ . Nevertheless, and to simplify the notation the constant 8π we have omitted 2 Γ F F it holds that Hxy = Hxy + Hyx , and we have proved using Fourier 8π

analysis that Hxy has moments of any order. The following results have been proved in [255]: 2 F 1. All the terms in the above decomposition of H xy exists in L (Ω) for x = y. 1

2. If |x − y| → 0, then the terms behave as |x − y| H −1 , so they can be integrated with respect to ρ(dx)ρ(dy). 3. The bound H < 23 is sharp: For H = local time diverges.

2 3

the weighted self-intersection

Notes and comments [5.1] The fractional Brownian motion was ﬁrst introduced by Kolmogorov [171] and studied by Mandelbrot and Van Ness in [217], where a stochastic integral representation in terms of a standard Brownian motion was established. Hurst developed in [141] a statistical analysis of the yearly water run-oﬀs xn are the values of n successive yearly of Nile river. Suppose that x1 , . . . ,

n water run-oﬀs. Denote by Xn = k=1 xk the cumulative values. Then, k Xk − n Xn is the deviation of the cumulative value Xk corresponding to k successive years from the empirical means as calculated using data for n years. Consider the range of the amplitude of this deviation: k k Rn = max Xk − Xn − min Xk − Xn 1≤k≤n 1≤k≤n n n

5.4 Vortex ﬁlaments based on fBm

319

and the empirical mean deviation G H n 2 H1 Xn I Sn = . xk − n n k=1

Based on the records of observations of Nile ﬂows in 622-1469, Hurst disH n covered that R Sn behaves as cn , where H = 0.7. On the other hand, the partial sums x1 + · · · + xn have approximately the same distribution as nH x1 , where again H is a parameter larger than 12 . These facts lead to the conclusion that one cannot assume that x1 , . . . , xn are values of a sequence of independent and identically distributed random variables. Some alternative models are required in order to explain the empirical facts. One possibility is to assume that x1 , . . . , xn are values of the increments of a fractional Brownian motion. Motivated by these empirical observations, Mandelbrot has given the name of Hurst parameter to the parameter H of fBm. The fact that for H > 12 fBm is not a semimartingale has been ﬁrst proved by [198] (see also Example 4.9.2 in Liptser and Shiryaev [201]). Rogers in [296] has established this result for any H = 12 . [5.2] Diﬀerent approaches have been used in the literature in order to deﬁne stochastic integrals with respect to fBm. Lin [198] and Dai and T Heyde [73] have deﬁned a stochastic integral 0 φs dBs as limit in L2 of Riemann sums in the case H > 12 . This integral does not satisfy the propT erty E( 0 φs dBs ) = 0 and it gives rise to change of variable formulae of Stratonovich type. A new type of integral with zero mean deﬁned by means of Wick products was introduced by Duncan, Hu and Pasik-Duncan in [86], assuming H > 12 . This integral turns out to coincide with the divergence operator (see also Hu and Øksendal [140]). A construction of stochastic integrals with respect to fBm with parameter H ∈ (0, 1) by a regularization technique was developed by Carmona and Coutin in [58]. The integral is deﬁned as the limit of approximating integrals with respect to semimartingales obtained by smoothing the singularity of the kernel KH (t, s). The techniques of Malliavin Calculus are used in order to establish the existence of the integrals. The ideas of Carmona and Coutin were further developed by Al` os, Mazet and Nualart in the case 0 < H < 12 in [8]. The interpretation of the divergence operator as a stochastic integral was ¨ ¨ in [78]. A stochastic calculus for introduced by Decreusefont and Ustunel the divergence process has been developed by Al`os, Mazet and Nualart in [9], among others. A basic reference for the stochastic calculus with respect to the fBM is the recent monograph by Hu [139]. An Itˆ oˆ’s formula for H ∈ (0, 1) in the framework of white noise analysis has been established by Bender in [22].

320

5. Fractional Brownian motion

We refer to [124] and [123] for the stochastic calculus with respect to fBM based on symmetric integrals. The results on the stochastic calculus with respect to the fBm are based on the papers [11] (case H > 12 ), [7] and [66] (case H < 12 ). [5.3] form

In [202], Lyons considered deterministic integral equations of the t xt = x0 + σ(xs )dgs , 0

0 ≤ t ≤ T , where the g : [0, T ] → R is a continuous functions with bounded p-variation for some p ∈ [1, 2). This equation has a unique solution in the space of continuous functions of bounded p-variation if each component of g has a H¨older continuous derivative of order α > p − 1. Taking into account that fBm of Hurst parameter H has locally bounded p-variation paths for p > 1/H, the result proved in [202] can be applied to Equation (5.78) in the case σ(s, x) = σ(x), and b(s, x) = 0, provided the coeﬃcient σ has a H¨older continuous derivative of order α > H1 − 1. The rough path analysis developed by Lyons in the [203] is a powerful technique based on the notion of p-variation which permits to handle differential equations driven by irregular functions (see also the monograph [204] by Lyons and Qian). In [70] Coutin and Qian have established the existence of strong solutions and a Wong-Zakai type approximation limit for stochastic diﬀerential equations driven by a fractional Brownian motion with parameter H > 14 using the approach of rough path analysis. In [299] Ruzmaikina establishes an existence and uniqueness theorem for ordinary diﬀerential equations driven by a H¨ o¨lder continuous function using H¨ o¨lder norms. The generalized Stieltjes integral deﬁned in (5.64), based on the fractional integration by parts formula, was introduced by Z¨ ahle in [355]. In this paper, the author develops an approach to stochastic calculus based on the fractional calculus. As an application, in [356] the existence and uniqueness of solutions is proved for diﬀerential equations driven by a fractional Brownian motion with parameter H > 12 , in a small random interval, provided the diﬀusion coeﬃcient is a contraction in the space W2β,∞ , where β 1 2 < β < H. Here W2,∞ denotes the Besov-type space of bounded measurable functions f : [0, T ] → R such that T T 2 (f (t) − f (s)) dsdt < ∞. |t − s|2β+1 0 0 d

In [254] Nualart and Rascanu have established the existence and uniqueness of solution for Equation (5.78) using an a priori estimate based on the fractional integration by parts formula, following the approach of Z¨ ahle. [5.4] The results of this section have been proved by Nualart, Rovira and Tindel in [255].

6 Malliavin Calculus in ﬁnance

In this chapter we review some applications of Malliavin Calculus to mathematical ﬁnance. First we discuss a probabilistic method for numerical computations of price sensitivities (Greeks) based on the integration by parts formula. Then, we discuss the use of Clark-Ocone formula to ﬁnd hedging portfolios in the Black-Scholes model. Finally, the last section deals with the computation of additional expected utility for insider traders.

6.1 Black-Scholes model Consider a market consisting of one stock (risky asset) and one bond (riskless asset). The price process of the risky asset is assumed to be of the form St = S0 eHt , t ∈ [0, T ], with

t

(µs −

Ht = 0

σ 2s )ds + 2

t

σ s dW Ws ,

(6.1)

0

where W = {W Wt , t ∈ [0, T ]} is a Brownian motion deﬁned in a complete probability space (Ω, F, P ). We will denote by {F Ft , t ∈ [0, T ]} the ﬁltration generated by the Brownian motion and completed by the P -null sets. The mean rate of return µt and the volatility process σ t are supposed to be measurable and adapted processes satisfying the following integrability conditions

322

6. Malliavin Calculus in ﬁnance

T

T

|µt |dt < ∞,

σ 2t dt < ∞

0

0

almost surely. By Itˆˆo’s formula we obtain that St satisﬁes a linear stochastic diﬀerential equation: Wt . (6.2) dS St = µt St dt + σ t St dW The price of the bond Bt , t ∈ [0, T ], evolves according to the diﬀerential equation dBt = rt Bt dt, B0 = 1, where the interest rate process is a nonnegative measurable and adapted process satisfying the integrability condition T rt dt < ∞, 0

almost surely. That is, Bt = exp

t

rs ds .

0

Imagine an investor who starts with some initial endowment x ≥ 0 and invests in the assets described above. Let αt be the number of non-risky assets and β t the number of stocks owned by the investor at time t. The couple φt = (αt , β t ), t ∈ [0, T ], is called a portfolio or trading strategy, and we assume that αt and β t are measurable and adapted processes such that T T T |β t µt |dt < ∞, β 2t σ 2t dt < ∞, |αt |rt dt < ∞ (6.3) 0

0

0

almost surely. Then x = α0 + β 0 S0 , and the investor’s wealth at time t (also called the value of the portfolio) is Vt (φ) = αt Bt + β t St . The gain Gt (φ) made by the investor via the portfolio φ up to time t is given by t t αs dBs + β s dSs . Gt (φ) = 0

0

Notice that both integrals are well deﬁned thanks to condition (6.3). We say that the portfolio φ is self-ﬁnancing if there is no fresh investment and there is no consumption. This means that the value equals to the intial investment plus the gain: t t Vt (φ) = x + αs dBs + β s dSs . (6.4) 0

0

From now on we will consider only self-ﬁnancing portfolios.

6.1 Black-Scholes model

323

We introduce the discounted prices deﬁned by t t σ2 St = Bt−1 St = S0 exp σ s dW Ws . µs − rs − s ds + 2 0 0 Then, the discounted value of a portfolio will be Vt (φ) = Bt−1 Vt (φ) = αt + β t St .

(6.5)

Notice that dVt (φ) = −rt Bt−1 Vt (φ)dt + Bt−1 dV Vt (φ) −1 St = −rt β t St dt + Bt β t dS = β t dSt , that is Vt (φ)

t

= x+

β s dSs

0

= x+

t

(µs − rs )β s Ss ds +

0

t

σ s β s Ss dW Ws .

(6.6)

0

Equations (6.5) and (6.6) imply that the composition αt on non-risky assets in a self-ﬁnancing portfolio is determined by the initial value x and βt: αt

= Vt (φ) − β t St t = x+ β s dSs − β t St . 0

On the other hand, (6.6) implies that if µt = rt for t ∈ [0, T ], then the value process Vt (φ) of any self-ﬁnancing portfolio is a local martingale.

6.1.1 Arbitrage opportunities and martingale measures Deﬁnition 6.1.1 An arbitrage is a self-ﬁnancing strategy φ such that VT (φ) > 0) > 0. V0 (φ) = 0, VT (φ) ≥ 0 and P (V In general, we are interested in having models for the stock price process without opportunities of arbitrage. In the case of discrete time models, the absence of opportunities of arbitrage is equivalent to the existence of martingale measures: Deﬁnition 6.1.2 A probability measure Q on the σ-ﬁeld FT , which is equivalent to P , is called a martingale measure (or a non-risky probability measure) if the discounted price process {St , 0 ≤ t ≤ T } is a martingale in the probability space (Ω, FT , Q).

324

6. Malliavin Calculus in ﬁnance

In continuous time, the relation between the absence of opportunities of arbitrage and existence of martingale measures is more complex. Let us assume the following additional conditions: σt > 0 for all t ∈ [0, T ] and

T

θ2s ds < ∞ 0

almost surely, where θt =

µt −rt σt .

Then, we can deﬁne the process

t 1 t 2 θs dW Ws − θs ds , Zt = exp − 2 0 0 which is a positive local martingale. If E(Z ZT ) = 1, then, by Girsanov theorem (see Proposition 4.1.2), the process Z is a martingale and the measure Q deﬁned by dQ dP = ZT is a probability measure, equivalent to P , such that under Q the process t D Wt = Wt + θs ds 0

Dt the price is a Brownian motion. Notice that in terms of the process W process can be expressed as t t σ2 Ds , St = S0 exp (rs − s )ds + σ s dW 2 0 0 and the discounted prices form a local martingale: t 1 t 2 −1 D St = Bt St = S0 exp σ s d Ws − σ ds . 2 0 s 0 Moreover, the discounted value process of any self-ﬁnancing stragety is also a local martingale, because from (6.6) we obtain t t Ds . Vt (φ) = x + β s dSs = x + σ s β s Ss dW (6.7) 0

0

Condition that there are no arbitrage opportunities verify (6.7) implies 2 T ing EQ 0 σ s β s Ss ds < ∞. In fact, this condition implies that the discounted value process Vt (φ) is a martingale under Q. Then, using the martingale property we obtain EQ VT (φ) = V0 (φ) = 0,

6.1 Black-Scholes model

325

so VT (φ) = 0, Q-almost surely, which contradicts the fact that P ( VT (φ) > 0) > 0. A portfolio φ is said to be admissible for the intial endowment x ≥ 0 if its value process satisﬁes Vt (φ) ≥ 0, t ∈ [0, T ], almost surely. Then, there are no arbitrage opportunities in the class of admissible portfolios. In fact, such an arbitrage will have a discounted value process Vt (φ) which is a nonnegative local martingale. Thus, it is a supermartingale, and, hence, EQ VT (φ) ≤ V0 (φ) = 0, so VT (φ) = 0, Q-almost surely, which contradicts the fact that P ( VT (φ) > 0) > 0. Note that, if we assume that the process σ t is uniformly bounded, then the discounted price process {St , 0 ≤ t ≤ T } is a martingale under Q, and Q is a martingale measure in the sense of Deﬁnition 6.1.2.

6.1.2 Completeness and hedging A derivative is a contract on the risky asset that produces a payoﬀ H at maturity time T . The payoﬀ is, in general, an FT -measurable nonnegative random variable H. Example 6.1.1 European Call-Option with maturity T and exercise price K > 0: The buyer of this contract has the option to buy, at time T , one share of the stock at the speciﬁed price K. If ST ≤ K the contract is worthless to him and he does not exercise his option. If ST > K, the seller is forced to sell one share of the stock at the price K, and thus the buyer can make a proﬁt ST − K by selling then the share at its market price. As a consequence, this contract eﬀectively obligates the seller to a payment of H = (S ST − K)+ at time T . Example 6.1.2 European Put-Option with maturity T and exercise price K > 0: The buyer of this contract has the option to sell, at time T , one share of the stock at the speciﬁed price K. If ST ≥ K the contract is worthless to him and he does not exercise his option. If ST < K, the seller is forced to buy one share of the stock at the price K, and thus the buyer can make a proﬁt ST − K by buying ﬁrst the share at its market price. As a consequence, this contract eﬀectively obligates the seller to a payment of H = (K − ST )+ at time T . Example 6.1.3 Barrier Option: H = (S ST − K)+ 1{TTa ≤T } , for some a > K > 0, a > S0 , where Ta = inf{t ≥ 0 : St ≥ a}. This contract is similar to the European call-option with exercise price K and maturity T , except that now the stock price has to reach a certain barrier level a > max(K, S0 ) for the option to become activated.

326

6. Malliavin Calculus in ﬁnance

+ T Example 6.1.4 Assian Option: H = T1 0 St dt − K . This contract is similar to the European call-option with exercise price K and maturity T , except that now the average stock price is used in place of the terminal stock price. We shall say that a nonnegative FT -measurable payoﬀ H can be replicated if there exists a self-ﬁnancing portfolio φ such that VT (φ) = H. The following proposition asserts that any derivative satisfying E(BT−2 ZT2 H 2 ) < ∞ is replicable, that is, the Black and Scholes model is complete. Its proof is a consequence of the integral representation theorem (see Theorem 1.1.3). Proposition 6.1.1 Let H be a nonnegative FT -measurable random variable such that E(BT−2 ZT2 H 2 ) < ∞. Then, there exists a self-ﬁnancing portfolio φ such that VT (φ) = H. Proof: By the integral representation theorem (Theorem 1.1.3) there exists an adapted and measurable process u = {ut , t ∈ [0, T ]} such that T 2 E 0 us ds < ∞ and BT−1 ZT H = E BT−1 ZT H + Set Mt = E

BT−1 ZT H|F Ft

T

us dW Ws . 0

= E BT−1 ZT H +

t

us dW Ws .

(6.8)

0

Notice that BT ZT−1 MT = H. We claim that there exists a self-ﬁnancing portfolio φ such that Vt (φ) = H. Deﬁne the self-ﬁnancing portfolio φt = (αt , β t ) by Zt−1 (ut + Mt θt ) , σ t St

βt

=

αt

= Mt Zt−1 − β t St .

The discounted value of this portfolio is Vt (φ) = αt + β t St = Zt−1 Mt ,

(6.9)

so, its ﬁnal value will be VT (φ) = BT VT (φ) = BT ZT−1 MT = H. Let us show that this portfolio is self-ﬁnancing. By Itˆ o’s formula we have Wt + θ2t dt) d(Z Zt−1 ) = Zt−1 (θt dW

6.1 Black-Scholes model

327

and d(Z Zt−1 Mt ) = Zt−1 dM Mt + Mt d(Z Zt−1 ) + dM Mt d(Z Zt−1 ) = Zt−1 ut dW Wt + Mt Zt−1 (θt dW Wt + θ2t dt) + Zt−1 ut θt dt Dt = Zt−1 (ut + Mt θt ) dW Dt . = σ t St β t dW Hence, Dt + Bt Zt−1 Mt rt dt dV Vt (φ) = d(Bt Zt−1 Mt ) = Bt σ t St β t dW Wt + Bt σ t St β t θt dt + Bt (αt + β t St )rt dt = Bt σ t St β t dW Wt + St β t (µt − rt ) dt + β t St rt dt + Bt αt rt dt = St σ t β t dW = St σ t β t dW Wt + β t St µt dt + Bt αt rt dt = αt dBt + β t dS St . The price of a derivative with payoﬀ H at time t ≤ T is given by the value at time t of a portfolio which replicates H. Under the assumptions of Proposition 6.1.1, from (6.8) and (6.9) we deduce T Vt (φ) = Bt Zt−1 E BT−1 ZT H|F Ft = Zt−1 E(Z ZT e− t rs ds H|F Ft ).

Assume now E(Z ZT ) = 1 and let Q be given by Exercise 4.2.11 we can write Vt (φ) = EQ (e−

T t

In particular, V0 (φ) = EQ (e−

6.1.3

rs ds

T 0

dQ dP

= ZT . Then, using

H|F Ft ).

rs ds

H).

(6.10)

Black-Scholes formula

Suppose that the parameters σ t = σ, µt = µ and rt = r are constant. In that case we obtain that the dynamics of the stock price is described by a geometric Brownian motion: σ2 µ− St = S0 exp t + σW Wt . 2 Moreover, θt = θ =

µ−r σ ,

and

θ2 Zt = exp −θW Wt − t . 2

328

6. Malliavin Calculus in ﬁnance

Dt = Wt + θt is a Brownian motion under Q, with So, E(Z ZT ) = 1, and W dQ dP = ZT , on the time interval [0, T ]. This model is complete in the sense that any payoﬀ H ≥ 0 satisfying EQ (H 2 ) < ∞ is replicable. In this case, we simply apply the integral representation theorem to the random variable e−rT H ∈ L2 (Ω, FT , Q) with D . In this way we obtain respect to the Wiener process W T −rT −rT Ds , e H = EQ e H + u s dW 0

and the self-ﬁnancing replicating portfolio is given by

where

ut , σ St

βt

=

αt

= Mt − β t St ,

Ft = EQ e−rT H + Mt = EQ e−rT H|F

t

Ds . u s dW

0

Consider the particular case of an European option, that is, H = Φ(S ST ), where Φ is a measurable function with linear growth. The value of this derivative at time t will be ST )|F Ft Vt (φ) = EQ e−r(T −t) Φ(S 2 D D St er(T −t) eσ(WT −Wt )−σ /2(T −t) )|F Ft . = e−r(T −t) EQ Φ(S Hence, Vt = F (t, St ),

(6.11)

where

2 D D F (t, x) = e−r(T −t) EQ Φ(xer(T −t) eσ(WT −Wt )−σ /2(T −t) ) .

(6.12)

Under general hypotheses on Φ (for instance, if Φ has linear growth, is continuous and piece-wise diﬀerentiable) which include the cases Φ(x)

=

(x − K)+ ,

Φ(x)

=

(K − x)+ ,

the function F (t, x) is of class C 1,2 . Then, applying Ito’s ˆ formula to (6.11) we obtain t t ∂F ∂F Du + (u, Su )Su dW (u, Su )Su du σ r Vt (φ) = V0 (φ) + ∂x ∂x 0 0 t ∂F 1 t ∂2F (u, Su )du + (u, Su ) σ 2 Su2 du. + 2 0 ∂x2 0 ∂u

6.1 Black-Scholes model

329

On the other hand, we know that Vt (φ) has the representation t t Du + Vt (φ) = V0 (φ) + σβ u Su dW rV Vu du. 0

0

Comparing these expressions, and taking into account the uniqueness of the representation of an Itˆ oˆ process, we deduce the equations ∂F (t, St ), ∂x ∂2F ∂F 1 (t, St ) + σ 2 St2 2 (t, St ) rF (t, St ) = ∂t 2 ∂x ∂F (t, St ). +rS St ∂x The support of the probability distribution of the random variable St is [0, ∞). Therefore, the above equalities lead to the following partial diﬀerential equation for the function F (t, x), where 0 ≤ t ≤ T, x ≥ 0 βt

=

∂F 1 ∂2F ∂F (t, x) + rx (t, x) + (t, x) σ 2 x2 ∂t ∂x 2 ∂x2 F (T, x)

= rF (t, x), =

Φ(x).

The replicating portfolio is given by βt αt

∂F (t, St ), ∂x = e−rt (F (t, St ) − β t St ) . =

(6.13) (6.14)

Formula (6.12) can be written as 2 D D F (t, x) = e−r(T −t) EQ Φ(xer(T −t) eσ(WT −Wt )−σ /2(T −t) ) ∞ √ 2 σ2 −rτ 1 √ = e Φ(xerτ − 2 τ +σ τ y )e−y /2 dy, 2π −∞ where τ = T − t is the time to maturity. In the particular case of an European call-option with exercise price K and maturity T , Φ(x) = (x − K)+ , and we get ∞ + √ 2 σ2 1 F (t, x) = √ e−y /2 xe− 2 τ +σ τ y − Ke−rθ dy 2π −∞ = xΦ(d+ ) − Ke−r(T −t) Φ(d− ), where log d−

= log

d+

=

x K

x K

+ r− √ σ τ + r+ √ σ τ

σ2 2

σ2 2

(6.15) τ , τ .

330

6. Malliavin Calculus in ﬁnance

Exercises 6.1.1 Consider the Black-Scholes model with constant volatility, mean rate of return and interest rate. Compute the price at time t0 , 0 ≤ t0 < T of a derivative whose payoﬀ is: T (i) H = T1 0 St dt. 4/3 (ii) H = ST . 6.1.2 Using

ST − K)− = ST − K (S ST − K)+ − (S

(6.16)

deduce a relation between the price of an European call-option and that of an European put-option. In particular, in the case of the Black-Scholes model with constant volatility, mean rate of return and interest rate, obtain a formula for the price of an European put-option from (6.15). 6.1.3 From (6.15) show that ∂F ∂x ∂2F ∂x2 ∂F ∂σ

=

Φ(d+ )

1 √ Φ (d+ ) xσ T √ = x T Φ (d+ ).

=

Deduce that F is a nondecreasing and convex function of x.

6.2 Integration by parts formulas and computation of Greeks In this section we will present a general integration by parts formula and we will apply it to the computation of Greeks. We will assume that the price process follows a Black-Scholes model with constant coeﬃcients σ, µ, and r. Let W = {W (h), h ∈ H} denote an isonormal Gaussian process associated with the Hilbert space H. We assume that W is deﬁned on a complete probability space (Ω, F, P ), and that F is generated by W . Proposition 6.2.1 Let F , G be two random variables such that F ∈ D1,2 . Consider an H-valued random variable u such that Du F = DF, uH = 0 a.s. and Gu(Du F )−1 ∈ Domδ. Then, for any continuously diﬀerentiable function function f with bounded derivative we have E(f (F )G) = E(f (F )H(F, G)), where H(F, G) = δ(Gu(Du F )−1 ).

6.2 Integration by parts formulas and computation of Greeks

Proof:

331

By the chain rule (see Proposition 1.2.3) we have Du (f (F )) = f (F )Du F.

Hence, by the duality relationship (1.42) we get E(f (F )G) = E Du (f (F ))(Du F )−1 G $ # = E D(f (F )), u(Du F )−1 G H = E(f (F )δ(Gu(Du F )−1 )).

This completes the proof. Remarks:

1. If the law of F is absolutely continuous, we can assume that the function f is Lipschitz. 2. Suppose that u is deterministic. Then, for Gu(Du F )−1 ∈ Domδ it suﬃces that G(Du F )−1 ∈ D1,2 . Suﬃcient conditions for this are given in Exercise 6.2.1. 3. Suppose we take u = DF . In this case GDF H(F, G) = δ , 2 DF H and Proposition 6.2.1 yields

E(f (F )G) = E

f (F )δ

GDF 2

DF H

.

(6.17)

A Greek is a derivative of a ﬁnancial quantity, usually an option price, with respect to any of the parameters of the model. This derivative is useful to measure the stability of this quantity under variations of the parameter. Consider an option with payoﬀ H such that EQ (H 2 ) < ∞. From (6.10) its price at time t = 0 is given by V0 = EQ (e−rT H). We are interested in computing the derivative of this expectation with respect to a parameter α, α being one of the parameters of the problem, Fα ). Then that is, S0 , σ, or r. Suppose that we can write H = f (F ∂V V0 dF Fα = e−rT EQ f (F Fα ) . (6.18) ∂α dα Using Proposition 6.2.1 we obtain ∂V V0 dF Fα = e−rT EQ f (F Fα )H Fα , . ∂α dα

(6.19)

In some cases the function f is not smooth and formula (6.19) provides better result in combination with Montecarlo simultation that (6.18). We are going to discuss several examples of this type.

332

6. Malliavin Calculus in ﬁnance

6.2.1

Computation of Greeks for European options

The most important Greek is the Delta, denoted by ∆, which by deﬁnition is the derivative of V0 with respect to the initial price of the stock S0 . Suppose that the payoﬀ H only depends on the price of the stock at the maturity time T . That is, H = Φ(S ST ). We call these derivative European options. From (6.13) it follows that ∆ coincides with the composition in ristky assets of the replicating portfolio. If Φ is a Lipschitz function we can write ∆=

∂S ST e−rT ∂V V0 = EQ (e−rT Φ (S ST ) )= EQ (Φ (S ST )S ST ). ∂S0 ∂S0 S0

Now we will apply Proposition 6.2.1 with u = 1, F = ST and G = ST . We have T Dt ST dt = σT ST . D u ST = 0

Hence, all the conditions appearing in Remark 2 above are satisﬁes in this case and we we have ⎛ −1 ⎞ T 1 WT ⎠=δ δ ⎝ ST . Dt ST dt = σT σT 0 As a consequence, ∆=

e−rT EQ (Φ(S ST )W WT ). S0 σT

(6.20)

The Gamma, denoted by Γ, is the second derivative of the option price with respect to S0 . As before we obtain 2 ∂S ST ∂ 2 V0 e−rT −rT Γ= = EQ e Φ (S ST ) EQ (Φ (S ST )S ST2 ). = 2 ∂S0 ∂S0 S02 Assuming that Φ is Lipschitz we obtain, taking G = ST2 , F = ST and u = 1 and applying Proposition 6.2.1 ⎛ −1 ⎞ T ⎠ = δ S T = S T WT − 1 Dt ST dt δ ⎝ST2 σT σT 0 and, as a consequence, WT EQ (Φ (S −1 . ST )S ST2 ) = EQ Φ (S ST )S ST σT

6.2 Integration by parts formulas and computation of Greeks

333

T Finally, applying again Proposition 6.2.1 with G = ST W σT − 1 , F = ST and u = 1 yields ⎛ −1 ⎞ T WT W 1 T ⎠ ⎝ −1 = δ δ ST Dt ST dt − σT σ2 T 2 σT 0 WT2 WT 1 − 2 − = σ2 T 2 σ T σT and, as a consequence, WT2 WT WT 1 −1 = EQ Φ(S − EQ Φ (S ST )S ST − ST ) . σT σ2 T 2 σ2 T σT Therefore, we obtain Γ=

2 WT 1 e−rT E − − W ) Φ(S S . Q T T S02 σT σT σ

(6.21)

The derivative with respect to the volatility is called Vega, and denoted by ϑ: ϑ=

∂S ST ∂V V0 = EQ (e−rT Φ (S ) = e−rT EQ (Φ (S ST ) ST )S ST (W WT − σT )). ∂σ ∂σ

Applying Proposition 6.2.1 with G = ST WT , F = ST and u = 1 yields ⎛ −1 ⎞ T ⎠ = δ WT − 1 δ ⎝ST (W WT − σT ) Dt ST dt σT 0 2 WT 1 = − − WT . σT σ As a consequence, 2 WT 1 − − WT ϑ = e−rT EQ Φ(S ST ) . σT σ

(6.22)

By means of an approximation procedure these formulas still hold although the function Φ and its derivative are not Lipschitz. We just need Φ to be piecewise continuous with jump discontinuities and with linear growth. In particular, we can apply these formulas to the case of and European call-option (Φ(x) = (x − K)+ ), and European put-option (Φ(x) = (K − x)+ ), or a digital option (Φ(x) = 1{x>K} ). We can compute the values of the previous derivatives with a Monte Carlo numerical procedure. We refer to [110] and [169] for a discussion of the numerical simulations.

334

6. Malliavin Calculus in ﬁnance

6.2.2 Computation of Greeks for exotic options Consider options whose payoﬀ is a function of the average of the stock price 1 T T 0 St dt, that is 1 T H=Φ St dt . T 0 For instance, an Asiatic call-option with exercise price K, is a derivative + T of this type, where H = T1 0 St dt − K . In this case there is no closed T formula for the density of the random variable T1 0 St dt. From (6.10) the price of this option at time t = 0 is given by 1 T −rT EQ Φ St dt . V0 = e T 0 Let us compute the Delta for this type of options. Set S T = We have ∆=

1 T

T 0

St dt.

∂S T e−rT ∂V V0 = EQ (e−rT Φ (S T ) )= EQ (Φ (S T )S T ). ∂S0 ∂S0 S0

We are going to apply Proposition 6.2.1 with G = S T , F = S T and ut = St . Let us compute 1 T σ T Dt F = Dt Sr dr = Sr dr, T 0 T t and

δ

T 0

GS·

=

St Dt F dt =

=

S· 2 δ T σ St dt ⎛ 0 ⎞ T T T S σS dr dt ⎟ t r Wt 0 t 2 ⎜ 0 St dW + ⎠ ⎝ T 2 T σ St dt S dt 0 t 0 T Wt 2 0 St dW + 1. T σ St dt 0

Notice that

0

Thus, δ

T 0

GS· St Dt F dt

T

1 St dW Wt = σ

ST − S0 − r

T

St dt . 0

2r 2 2 (S ST − S0 ) +1− 2 = 2 = T 2 σ σ σ 0 St dt

ST − S0 −m , T St dt 0

6.2 Integration by parts formulas and computation of Greeks

335

2

where m = r − σ2 . Finally, we obtain the following expression for the Delta: ST − S0 2e−rT ∆= EQ Φ S T −m . S0 σ 2 T ST For other type of path dependent options it is not convenient to take u = 1 in the integration by parts formula. Consider the general case of an option depending on the prices at a ﬁnite number of times, that is, H = Φ(S St1 , . . . , Stm ), where Φ : Rm → R is a continuously diﬀerentiable function with bounded partial derivatives and 0 < t1 < t2 < · · · < tm = T . We introduce the set Γm deﬁned by tj 2 Γm = a ∈ L ([0, T ]) : at dt = 1 ∀j ∀ = 1, . . . , m . 0

Then we have ∆ = EQ

T

at dW Wt

H

.

0

In fact, we have a

D H

=

m

∂j Φ(S St1 , . . . , Stm )Da Stj

j=1 m

= σ

∂j Φ(S St1 , . . . , Stm )S Stj .

j=1

As a consequence, m ∂H 1 Da H = ∂j Φ(S St1 , . . . , Stm )S Stj = ∂S0 S0 j=1 σS0

and we obtain −rT

∆=e

EQ

∂H ∂S0

e−rT e−rT EQ (Da H) = EQ = σS0 σS0

We can take for instance a =

1 t1 1[0,t1 ]

T

at dW Wt

H

.

0

and we get

e−rT Wt1 ∆= EQ H . σS0 t1

(6.23)

Formula (6.23) is not very useful for simulation due to the inestability of Wt1 t1 if t1 is small. For this reason, speciﬁc alternative methods should be

336

6. Malliavin Calculus in ﬁnance

used to handle every case. For example, in [169] the authors consider an up in and down call option with payoﬀ H = 1{inf i=1,...,m Sti ≤D} 1{supi=1,...,m Sti ≥U } 1{ST
It is proved in [169] that if Ψ : [0, ∞) → [0, 1] is a function in Cb∞ such that Ψ(x) = 1 if x ≤ a/2 and Ψ(x) = 0 if x > a, where U > S0 + a2 > S0 − a2 > D, then, Ψ(Y Y· ) S0 e−rT EQ H δ T . ∆= σ Ψ(Y Yt )dt 0

Exercises 6.2.1 Suppose that G ∈ D1,4 , F ∈ D2,2 and u is an H-valued random 6 variable such that: E(G6 ) < ∞, E((Du F )−12 ) < ∞, and E(DDu F H ) < ∞. Show that G(Du F )−1 ∈ D1,2 . 6.2.2 Using formulas (6.20), (6.21), and (6.22) compute the values of ∆, Γ and ϑ for an European call option with exercise price K and compare the results with those obtained in Exercise 6.1.3. 6.2.3 Compute ∆, Γ and ϑ for a digital option using formulas (6.20), (6.21), and (6.22).

6.3 Application of the Clark-Ocone formula in hedging In this section we discuss the application of Clark-Ocone formula to ﬁnd explicit formulas for a replicating portfolio in the Black-Scholes model.

6.3.1 A generalized Clark-Ocone formula Suppose that Dt = Wt + W

t

θs ds, 0

where θ = {θt , t ∈ [0, T ]} is an adapted and measurable process such that T 2 θt dt < ∞ almost surely. Suppose that E(Z ZT ) = 1, where the process 0

6.3 Application of the Clark-Ocone formula in hedging

337

Zt is given by t 1 t 2 Zt = exp − θs dW Ws − θs ds . 2 0 0 D = Then, by Girsanov Theorem (see Proposition 4.1.2), the process W D {Wt , t ∈ [0, T ]} is a Brownian motion under the probability Q on FT given by dQ dP = ZT . The Clark-Ocone formula established in Proposition 1.3.14 can be generalized in oder to represent an FT -measurable random variable F as a D . Notice that, in general, stochastic integral with respect to the process W D D W W Ft , 0 ≤ t ≤ T } denotes the family of σwe have FT ⊂ FT (where {F D D ﬁelds generated by W ) and usually FTW = FT . Thus, an FT -measurable D random variable F may not be FTW -measurable and we cannot obtain a D simply by applying representation of F as an integral with respect to W D on the probaiblity the Clark-Ocone formula to the Brownian motion W D W space (Ω, FT , Q). In order to establish a generalized Clark-Ocone formula we need the following technical lemma. Its proof is left as an exercise (Exercise 6.3.1). Lemma 6.3.1 Let F be an FT -measurable random variable such that F ∈ D1,2 and let θ ∈ L1,2 . Assume T (i) E(Z ZT2 F 2 ) + E(Z ZT2 0 (Dt F )2 dt) < ∞, 2 T T 2 2 T Ws + t θs Dt θs ds dt < ∞. (ii) E ZT F 0 θt + t Dt θs dW Then ZT F ∈ D1,2 and

Dt (Z ZT F ) = ZT Dt F − ZT F

θt +

T

Dt θs dW Ws + t

T

θs Dt θs ds . t

Theorem 6.3.1 Let F be an FT -measurable random variable such that F ∈ D1,2 and let θ ∈ L1,2 . Suppose that conditions (i) and (ii) of Lemma 6.3.1 hold. Then T T Ds |F Dt . EQ Dt F − F Dt θs dW Ft dW F = EQ (F ) + 0

Proof:

t

Put Yt = EQ (F |F Ft ). Using Exercise 4.2.11 we can write Yt = Zt−1 E(Z ZT F |F Ft ),

where Zt−1 = exp

t

θs dW Ws + 0

1 2

0

t

θ2s ds .

338

6. Malliavin Calculus in ﬁnance

Then, by Clark-Ocone formula, we have t E(Z ZT F |F Ft ) = E(Z ZT F ) + E(Ds (Z ZT F )|F Fs )dW Ws . 0

Hence, Yt = Zt−1 EQ (F ) + Zt−1

t

E(Ds (Z ZT F )|F Fs )dW Ws .

(6.24)

0

From Lemma 6.3.1 we obtain ZT F )|F Ft ) E(Dt (Z

= E

Dt F − F

ZT

= Zt EQ

T

θt +

T

θt +

Ds Dt θs dW

t

Dt F − F

|F Ft

Ds Dt θs dW

|F Ft

t

= Zt Ψt − Zt Yt θt , where

Ψt = EQ (Dt F − F

(6.25) T

Ds |F Dt θs dW Ft ).

t

Substituting (6.25) into (6.24) yields Yt = Zt−1 EQ (F ) + Zt−1

t

Zs Ψs dW Ws − Zt−1

0

t

Zs Ys θs dW Ws . 0

Applying Itˆ o’s formula and using d(Z Zt−1 ) = Zt−1 (θt dW Wt + θ2t dt) we get dY Yt

= Yt (θt dW Wt + θ2t dt) + Ψt dW Wt − Yt θt dW Wt + θt Ψt dt − Yt θ2t dt Dt . = Ψt dW

This completes the proof.

6.3.2 Application to ﬁnance Let H be the payoﬀ of a derivative in the Black-Scholes model (6.2). Suppose that E(BT−2 ZT2 H 2 ) < ∞. We have seen that H is replicable, as a consequence of the integral representation theorem. Furthermore, if φt = (αt , β t ) is a replicating portfolio, we have seen that t Ds . Vt (φ) = EQ (BT−1 H) + σ s Ss β s dW 0

6.3 Application of the Clark-Ocone formula in hedging

339

Suppose now that BT−1 H ∈ D1,2 , θ ∈ L1,2 and the conditions (i) and (ii) of Lemma 6.3.1 are satisﬁed by F = BT−1 H and θ. Then, we conclude that T −1 −1 Ds |F Dt θs dW Ft . σ t St β t = EQ Dt B H − B H T

T

t

Hence, Bt βt = EQ σ t St

Dt BT−1 H − BT−1 H

T

Ds |F Dt θs dW Ft

.

t

If the interest rate rt = r is constant this formula reduces to T e−r(T −t) Ds |F βt = EQ Dt H − H Dt θs dW Ft . σ t St t In the particular case µt = µ, σ t = σ, we obtain Dt θs = 0, and βt =

e−r(T −t) EQ (Dt H|F Ft ) . σS St D

In that case, the only hypothesis required is H ∈ D1,2,W . In fact, it suﬃces D to apply the ordinary Clark-Ocone formula for the Brownian motion W ert ut and use that β t = σSt , where e−rT H = EQ e−rT H +

T

Dt . u t dW

0

Consider the particular case of an European option with payoﬀ H = Φ(S ST ). Then βt

e−r(T −t) EQ (Φ (S ST )σS ST |F Ft ) σS St ST −r(T −t) ST = e EQ Φ ( St ) |F Ft St St

=

= e−r(T −t) EQ (Φ (xS ST −t )S ST −t ) |x=St . In this way we recover the fact that β t coincides with ∂F ∂x (t, St ), where F (t, x) is the price function. Consider now an option whose payoﬀ is a function of the average of the T stock price S T = T1 0 St dt, that is H = Φ S T . In this case we obtain eT −t 1 T βt = EQ Φ (S T ) Sr dr|F Ft . St T t

340

6. Malliavin Calculus in ﬁnance

We can write

where S t =

1 t

t 0

t 1 ST = St + T T

T

Sr dr, t

Sr dr. As a consequence we obtain

tx y(T − t) y(T − t) e−r(T −t) + EQ Φ S T −t S T −t |x=S t ,y=St . βt = St T T T

Exercises 6.3.1 Prove Lemma 6.3.1. Dt = Wt + t θs ds, where θ = {θt , t ∈ [0, T ]} is an adapted and 6.3.2 Let W 0 measurable process. Use the generalized Clark-Ocone formula to ﬁnd the D of the following integral representation in terms of the Wiener process W random variables: (i) F = WT2 , and θ is bounded and belongs to D1,p for some p > 2, (ii) F = exp(aW WT ), and θ is bounded and belongs to D1,p for some p > 2, (iii) F = exp(aW WT ) and θt = Wt . 6.3.3 Consider the Black-Scholes model with constant volatility, mean rate of return and interest rate. Consider the particular case of an European option with payoﬀ H = exp(aW WT ). Find a replicating portfolio using the Clark-Ocone formula.

6.4 Insider trading Suppose that in a ﬁnancial market there are two types of agents: a natural agent whose information coincides with the natural ﬁltration of the price process, and an insider who possesses some extra information from the beginning of the trading interval [0, T ]. The simpler modelization of this additional information consists in assume the knowledge of a given random variable L. Two important questions in this context are: i) How can one calculate the additional utility of the insider? ii) Does the insider have arbitrage opportunities? Consider the Black-Scholes model for the price process St with a measurT able and adapted mean rate of return µt satisfying E 0 |µt |dt < ∞ and T a measurable and adapted volatility process satisfying E 0 σ 2t dt < ∞ and σ t > 0 a.s. The price process is then given by Wt ). dS St = St (µt dt + σ t dW

6.4 Insider trading

341

As usual, we denote, {F Ft , t ≥ 0} the ﬁltration generated by the Wiener process W and the P -null sets. The price of the bond is given by

t

Bt = exp

rs ds , 0

where rt is a non negative measurable and adapted process satisfying T

E

rt dt

< ∞.

0

A natural insider will use a self-ﬁnancing portfolio φt = (αt , β t ) where the processes αt and β t are measurable and Ft -adapted, satisfying

T

0

T

|β t µt |dt < ∞,

T

|αt | rt dt < ∞, 0

β 2t σ 2t dt < ∞

(6.26)

0

almost surely. That is, the value process Vt (φ) = αt Bt + β t St satisﬁes the self-ﬁnancing condition St . dV Vt (φ) = rt αt Bt dt + β t dS

(6.27)

We restrict ourselves to strictly admissible portfolios, that is to portfolios St is the φ satisfying Vt (φ) > 0 for all t ∈ [0, T ]. The quantity π t = Vβtt(φ) proportion of the wealth invested in stocks, and it determines the portfolio. In terms of π t the value process (denoted by Vt ) satisﬁes the following linear stochastic diﬀerential equation Vt dt + π t σ t Vt dW Wt . dV Vt = (ρt + (µt − rt )π t )V The solution of this equation is ( t

Vt = V0 Bt exp 0

2

(π s σ s ) (µs − rs )π s − 2

)

ds +

t

π s σ s dW Ws

. (6.28)

0

One of the possible objectives of an investor is to maximize the expected utility from terminal wealth, by means of choosing an appropriate portfolio. We will consider the logarithmic utility function u(x) = log x to measure the utility that the trader draws from his wealth at the maturity time T . Then, the problem is to ﬁnd a portfolio π t which maximizes the expected utility: (6.29) ΦF = max E (log VT ) , π

342

6. Malliavin Calculus in ﬁnance

where

T

E (log VT ) = log V0 + E +E 0

T

(

rs ds 0

(π s σ s ) (µs − rs )π s − 2

2

)

T

π s σ s dW Ws

ds +

.

0

t Due to the local martingale property of the stochastic integral 0 π s σ s dW Ws , the utility maximization probllem reduces to ﬁnd the maximum of ( ) 2 T (π s σ s ) E (µs − rs )π s − ds . 2 0 We can write this expression as 2 T T (µs − rs )2 µs − rs 1 ds + E ds , πs σs − − E 2 σs 2σ 2s 0 0 and the solution to the maximization problem (6.29) is (Merton’s formula) πs =

and ΦF = E

0

T

µs − rs , σ 2s

(6.30)

(µs − rs )2 ds . 2σ 2s

Consider now the problem of an insider trader. Assume that the investor is allowed to use a portfolio φt = (αt , β t ) which is measurable and Gt -adapted, where {Gt , t ∈ [0, T ]} is a ﬁltration which is modelling the information of the investor. In principle no assumption will be made on this ﬁltration. In the particular case where the additional information is given by a random variable L which is FT -measurable (or more generally, FT +ε -measurable for some ε > 0) we have Gt = Ft ∨ σ(L). We also assume condition (6.26). Now the self-ﬁnancing condition (6.27), by deﬁnition, will be t t t Vt (φ) = V0 + rs αs Bs ds + β s µs Ss ds + β s σ s Ss d− Ws , (6.31) 0

t

0

0

where 0 β s Ss σ s d− Ws denotes the forward stochastic integral introduced in Deﬁnition 3.2.1, and the process β t σ t St satisﬁes the assumptions of

6.4 Insider trading

343

Proposition 3.2.3. That is, the mapping t → β t σ t St is continuous from 1,2 . [0, T ] into D1,2 , and βσS ∈ L1− St as before. Then, Assume Vt (φ) > 0 for all t ∈ [0, T ] and set π t = Vβtt(φ) from (6.31) we obtain dV Vt = rt Vt + (µt − rt ) π t Vt dt + π t σ t Vt d− Wt .

(6.32)

Under some technical conditions on the processes π t , σ t and µt (see Exercise 6.4.1), the process Vt deﬁned by ) ( t 2 t T (π s σ s ) r ds − s ds + Vt = V0 e 0 exp (µs − rs ) π s − π s σ s d Ws 2 0 0 (6.33) satisﬁes the linear stochastic diﬀerential equation (6.32). Let us denote by AG the class of portfolios π = {π t , t ∈ [0, T ]} satisfying the following conditions: (i) π is Gt -adapted. (ii) The processes π and σ are continuous in the norm of D1,4 and π ∈ 1,2 . L2− T (iii) E 0 |(µs − ρs ) π s | ds < ∞. We need the following technical lemma. Lemma 6.4.1 Let π and σ be measurable process which are continuous 1,2 in the norm of D1,4 and π ∈ L2− . Suppose that σ is Ft -adapted. Then the T − forward integral 0 π s σ s d Ws exists and

T

π s σ s d − Ws =

0

Moreover,

T 0

T

T

π s σ s dW Ws − 0

D− π

0

s

σ s ds.

π s σ s d− Ws equals to the following limit in L1 (Ω)

T

−

π s σ s d Ws = lim 0

|π|↓0

n−1 i=0

π ti

ti+1

σ s dW Ws .

ti

1,2 and (D− (πσ))t = (D− π)t σ t . This Proof: Notice ﬁrst that πσ ∈ L2− follows from the fact that for any s > t we have

Ds (π t σ t ) = σ t Ds π t . The proof of the lemma can be done using the same ideas as in the proof of Proposition 3.2.3 (Exercise 6.4.2).

344

6. Malliavin Calculus in ﬁnance

We aim to solve the optimization problem ΦG = max E (log VTπ ) .

(6.34)

π∈AG

The following theorem provides a characterization of optimal portfolios. Theorem 6.4.1 The following conditions are equivalent for a portfolio π ∗ ∈ AG : (i) The portfolio π ∗ is optimal for problem (6.34). s Wr |Gt , is absolutely continuous in [t, T ] (ii) The function s → E 0 σ r dW for any t ∈ [0, T ) and there exists s d 2 ∗ E µs − ρs − σ s π s |Gt = − E σ r dW Wr |Gt (6.35) ds 0 for almost all s ∈ [t, T ]. Proof:

Set J(π)

= E

(log VTπ )

T

(

= E 0

+E

T

T

− V0 − E

rs ds 0 2

(π s σ s ) (µs − rs ) π s − 2 π s σ s d − Ws

) ds

.

(6.36)

0

Assume ﬁrst that π ∗ ∈ AG is optimal. We have J(π ∗ ) ≥ J(π ∗ + εβ) for any β ∈ AG and ε ∈ R. Therefore d J(π ∗ + εβ)|ε=0 = 0. dε Hence,

T

E

µs − rs − σ 2s π ∗s β s ds +

0

T

−

β s σ s d Ws

= 0,

(6.37)

0

for all β ∈ AG . In particular, applying this to β u = G1(s,t] (u), where 0 ≤ r < t ≤ T and G is Gt -measurable and bounded, we obtain E r

t

µs − ρs − σ 2s π ∗s ds +

t

σ s dW Ws |Gr r

= 0,

(6.38)

6.4 Insider trading

345

which implies (ii). Conversely, integrating (6.35) we obtain (6.38). For any β ∈ AG we have ti+1 n−1 T − β s d Xs = lim E β ti ( σ s dW Ws ) E 0

|π|↓0

ti

i=0

n−1 = lim E β ti E |π|↓0

= lim

|π|↓0

= lim

|π|↓0

i=0 n−1 i=0 n−1

ti

E β ti E

ti+1

E β ti

= E

T

ti

ti+1

σ s dW Ws |Gti µs − rs − σ 2s π ∗s ds|Gti

µs − rs − σ 2s π ∗s ds

ti

i=0

ti+1

∗

µs − rs − σ 2s π s β s ds

0

and (6.37) holds. This implies that the directional derivative of J at π ∗ with respect to the direction β denoted by Dβ J(π ∗ ) is zero. Note that J : AG → R is concave. Therefore, for all α, β ∈ AG and ε ∈ (0, 1), we have α + εβ) − J(α) 1−ε α ≥ (1 − ε)J( ) + εJ(β) − J(α) 1−ε α α = J( ) − J(α) + ε J(β) − J( ) . 1−ε 1−ε

J(α + εβ) − J(α) = J((1 − ε)

1 Now, with 1−ε = 1 + η we have 1 1+η α lim ) − J(α) = lim (J(α + ηα) − J(α)) = Dα J(α), J( ε→0 ε η→0 1−ε η

and we obtain Dβ J(α) = lim

ε→0

1 (J(α + εβ) − J(α)) ≥ Dα J(α) + J(β) − J(α). ε

In particular, applying this to α = π ∗ and using that Dβ J(π ∗ ) = 0 we get J(β) − J(π ∗ ) ≤ 0 for all β ∈ AG and this proves that π ∗ is optimal. The characterization theorem provides a closed formula for the optimal portfolio π ∗ . In fact, we have π ∗t E(σ 2t |Gt ) = E(µt − rt |Gt ) + a(t),

(6.39)

346

6. Malliavin Calculus in ﬁnance

where 1 a(t) = lim E h↓0 h

t+h

σ r dW Wr |Gt

.

t

Note that the optimal portfolio has a similar form as the solution of the Merton problem (6.30). We compute now the value of the optimal portfolio when it exists. From (6.39) we have E(ν t |Gt ) + a(t) , (6.40) π ∗t = E(σ 2t |Gt ) where ν t = µt − rt . Substituting (6.40) into (6.36) yields ( 2 ) T E(ν s |Gs ) + a(s) σ 2s E(ν s |Gs ) + a(s) ∗ − νs ds J(π ) = E E(σ 2s |Gs ) 2 E(σ 2s |Gs ) 0 T E(ν s |Gs ) + a(s) σ s d − Ws . +E E(σ 2s |Gs ) 0 Now, using the properties of the conditional expectation and applying Proposition 3.2.3 we get ( ) 2 T a(s)2 1 (E(µs − rs |Gs )) ∗ J(π ) = E − ds 2 E(σ 2s |Gs ) E(σ 2s |Gs ) 0 T − E(µ· − r· |G· ) + a(·) +E σ · ds . (6.41) D E(σ 2· |G· ) 0 s Example 1 (Partial observation case). Assume Gt ⊂ Ft . Then, s d E σ r dW Wr |Gt = 0 ds 0 for all s > t. That is a(t) = 0, and the optimal portfolio is given by π ∗t =

E(µt − rt |Gt ) E(σ 2t |Gt )

if the right-hand side is well deﬁned as an element of AG . In this case, the optimal utility is ( ) 2 T 1 (E(µs − rs |Gs )) ∗ J(π ) = E ds . 2 E(σ 2s |Gs ) 0 Example 2 (Insider strategy). Suppose Ft ⊂ Gt . Then π ∗t =

µt − rt a(t) + 2 2 σt σt

6.4 Insider trading

and 1 J(π ∗ ) = E 2

(

T

0

347

) 2 T (D− a)s (µs − rs ) a(s)2 − ds + E ds . σ 2s σ 2s σs 0 (6.42)

Consider the particular case where the σ-ﬁeld Gt is generated by an FT measurable random variable L. We can apply the approach of the enlargement of ﬁltrations and deduce that W is a semimartingale with respect to the ﬁltration Gt . Suppose that there exists a Gt -progresively measurable T process µGt such that 0 |µGt |dt < ∞ almost surely and Dt = Wt − W

t

µGs ds

0

is a Gt -Brownian for any Gt -progresively measurable portfolio motion. Then, T π such that E 0 π 2s σ 2s ds < ∞ we can write E

T

−

π s σ s d Ws

T

=E

0

π s σ s µGs ds

0

and, as a consequence, ( ) 2 T σ ) (π s s J(π) = E ds µs − rs + σ s µGs π s − 2 0 2 2 1 T µs − rs 1 T µs − rs G G = − + µs − σ s π s ds + + µs ds. 2 0 σs 2 0 σs Thus, the optimal portfolio is given by π ∗t =

µs − rs µG + s. 2 σs σs

On the other hand, the additional expected logarithmic utility will be 2 2 T T µs − rs µs − rs 1 1 G E + µs ds − E ds 2 σs 2 σs 0 0 T 1 G 2 E µs ds . = 2 0 because T T µs − rs µs − rs G Ds dW W s − dW E µs ds = E = 0, σs σs 0 0

348

6. Malliavin Calculus in ﬁnance

provided we assume E

a(t)

=

1 lim E h↓0 h

=

σ t µGt .

T µs −rs 2 0

t+h

σs

ds

Ds |Gt σ s dW

t

< ∞. Notice that 1 + lim E h↓0 h

t+h

σ s µGs ds|Gt

t

Example 6.4.1 Suppose that L = WS , where S ≥ T . Then µGt =

WS − W t , S−t

and the additional expected utillity is inﬁnite because 2 T T WS − W t S dt = log , E dt = S−t S−T 0 0 S−t which is ﬁnite if and only if S > T .

Exercises 2

2,4 1,2 1,2 6.4.1 Suppose that πσ ∈ L−loc , V0 ∈ Dloc , (µt − ρt )π t − (πt 2σt ) ∈ Lloc and the value process Vt has continuous trajectories. Then, appy the Itˆ oˆ’s formula (3.36) in order to deduce (6.32).

6.4.2 Complete the proof of Lemma 6.4.1. 6.4.3 Compute the additional expected logarithmic utility when L = 1{St0 ≥K} , where t0 ≥ T .

Notes and comments [6.1] The Black-Sholes formula for option pricing was derived by Black and Scholes in a paper published in 1973 [44]. Their results were inﬂuenced by the research developed by Samuelson and Merton. There are many reference books on mathematical ﬁnance where the techniques of stochastic calculus are applied to derive the basic formulas for pricing and hedging of derivatives (see Karatzas [160], Karatzas and Shreve [165], Lamberton and Lapeyre [189], and Shreve [312]). The recent monograph by Malliavin and Thalmaier [216] discusses a variety of applications of Malliavin calculus in mathematical ﬁnance. [6.2] We refer to [169] as a basic expository paper on the applications of the integration by parts formula of Malliavin calculus to Monte Carlo simulations of greeks. In [110] and [111] the Malliavin calculus is applied to

6.4 Insider trading

349

derive formulas for the derivative of the expectation of a diﬀusion process with respect to the parameters of the equation. The results on the case of an option depending on the price at a ﬁnite number have been taken from [122] and [31]. [6.3] The generalized Clark-Ocone formula and its applications in hedging in the multidimensional case has been studied by Karatzas and Ocone in [161]. [6.4] A pioneering paper in the study of the additional utility for insider traders is the work by Karatzas and Pikovsky [163]. They assume that the extra information is hidden in a random variable L from the beginning of the trading interval, and they make use of the technique of enlargement of ﬁltrations. For another work on this topic using the technique of enlargement of ﬁltrations we refer to [12]. The existence of opportunities of arbitrage for the insider has been investigated in [127] and [151]. For a review of the role of Malliavin calculus in the problem of insider trading we refer to the work by Imkeller [149]. The approach of anticipating stochastic calculus in the problem of insider trading has been developed in [196].

A Appendix

A.1 A Gaussian formula

∞ Let {an , n ≥ 1} be a sequence of real numbers such that n=1 a2n < ∞. Suppose that {ξ n , n ≥ 1} is a sequence of independent random variables deﬁned in the probability space (Ω, F, P ) with distribution N (0, 1). Then, for each p > 0 p ∞ p2 ∞ an ξ n (t) = Ap a2n , E n=1

where Ap =

(A.1)

n=1

R

x2 |x|p √ e− 2 dx. 2π

A.2 Martingale inequalities Let {M Mt , t ∈ [0, T ]} be a continuous local martingale with respect to an Ft , t ≥ 0}. The following inequalities play a increasing family of σ-ﬁelds {F fundamental role in the stochastic calculus:

352

A.3 Continuity criteria

Doob’s maximal inequality For any p > 1 we have p p E sup |M Mt |p ≤ E(|M MT |p ). p−1 0≤t≤T

(A.2)

Actually (A.2) holds if we replace |M Mt | by any continuous nonnegative submartingale. Burkholder-Davis-Gundy inequality For any p > 0 there exist constants c1 (p) and c2 (p) such that p p c1 (p)E M T2 ≤ E( sup |M Mt |p ) ≤ c2 (p)E M T2 .

(A.3)

0≤t≤T

This inequality still holds if we replace T by a stopping time S : Ω → [0, T ]. On the other hand, applying the Gaussian formula (A.1) and (A.3) one can show Burkholder’s inequality for Hilbert-valued martingales. That is, if {M Mt , t ∈ [0, T ]} is a continuous local martingale with values in a separable Hilbert space K, then for any p > 0 one has p (A.4) E (M MT p ) ≤ cp E M T2 , where M T =

∞

M, ei K T ,

i=1

{ei , i ≥ 1} being a complete orthonormal system in K. Exponential inequality For any δ > 0 and ρ > 0 we have 2 δ P M T < ρ, sup |M Mt | ≥ δ ≤ 2 exp − . 2ρ 0≤t≤T

(A.5)

Two-parameter martingale inequalities Let W = {W (z), z ∈ R2+ } be a two-parameter Wiener process. Consider the σ-ﬁelds Fz Fs1

= σ{W (z ), z ≤ z}, = σ{W (s , t ), 0 ≤ s ≤ s, t ≥ 0},

Ft2

= σ{W (s , t ), 0 ≤ t ≤ t, s ≥ 0}.

A.3 Continuity criteria

353

We will say that a random ﬁeld ξ = {ξ(z), z ∈ R2+ } is adapted if ξ(z) is Fz measurable for all z. Also, we will say that ξ is 1-adapted (resp. 2-adapted) if ξ(s, t) is Fs1 -measurable (resp. Ft2 -measurable). Let ξ = {ξ(z), z ∈ R2+ } be a measurable and adapted process such that S

T

ξ 2 (z)dz

E 0

<∞

0

for all (S, T ). Deﬁne, for s ≤ S, t ≤ T ,

s

t

ξ(z)W (dz).

M (s, t) = 0

(A.6)

0

Applying Doob’s inequality (A.2) (see [55]) twice, we obtain for any p > 1 ⎞ ⎛ p 2p S T p ⎜ p⎟ E ξ(z)W (dz) . (A.7) E ⎝ sup |M (s, t)| ⎠ ≤ 0 0 p−1 0≤s≤S 0≤t≤T

Moreover, if ξ is 1-adapted (resp. 2-adapted), the stochastic integral (A.6) can also be deﬁned, and it is a continuous martingale with respect to the coordinate s (resp. t), whose quadratic variation is given by

s

M (s, t) =

t

ξ(z)2 dz. 0

0

As a consequence, (A.3) and (A.7) yield, for any p > 1, p2 s t s t 2 c1 (p)E ξ(z) dz ≤ E(| ξ(z)W (dz)|p ) 0 0 0 0 p2 s t 2 ≤ c2 (p)E ξ(z) dz . (A.8) 0

0

A.3 Continuity criteria The real analysis lemma due to Garsia, Rodemich, and Rumsey’s [114] is a powerful tool to deduce results on the modulus of continuity of stochastic processes from estimates of the moments of their increments. The following version of this lemma has been taken from Stroock and Varadhan [322, p. 60]. Lemma A.3.1 Let p, Ψ : R+ → R+ be continuous and strictly increasing functions vanishing at zero and such that limt↑∞ Ψ(t) = ∞. Suppose that

354

A.3 Continuity criteria

φ : Rd → E is a continuous function with values on a separable Banach space (E, · ). Denote by B the open ball in Rd centered at x0 with radius r. Then, provided φ(t) − φ(s) Ψ Γ= dsdt < ∞, p(|t − s|) B B it holds, for all s, t ∈ B,

2|t−s|

φ(t) − φ(s) ≤ 8

−1

Ψ 0

4d+1 Γ λd u2d

p(du),

where λd is a universal constant depending only on d. Now suppose that X = {X(t), t ∈ Rd } is a continuous stochastic process with values on a separable Banach space (E, · ) such that the following estimate holds: (A.9) E(X(t) − X(s)γ ) ≤ H|t − s|α for some H > 0, γ > 0, α > d, and for all s, t ∈ B. m+2d Then taking Ψ(x) = xγ and p(x) = x γ , with 0 < m < α − d, one obtains X(t) − X(s)γ ≤ Cd,γ,m |t − s|m Γ, (A.10) X(t)−X(s) γ for all s, t ∈ B, where Γ = B B |t−s|m+2d dsd. Moreover, if E(X(t0 )γ ) < ∞ for some t0 ∈ E, then we can conclude that E( sup X(t)γ ) < ∞

(A.11)

|t|≤a

for any a > 0. If X is not supposed to be continuous, one can show by an approximation argument that (A.9) implies the existence of a continuous version of X satisfying (A.10) (see [315]). This proves the classical Kolmogorov continuity criterion. Actually, (A.10) follows from E(Γ) < ∞ if we assume γ ≥ 1, as it can be proved by an approximation of the identify argument.

A.4 Carleman-Fredholm determinant Let (T, B, µ) be a measure space, and let K ∈ L2 (T × T ). Assume that the ≥ 1} be a complete Hilbert space H = L2 (T ) is separable and let {ei , i

n orthonormal system in H. In the particular case K = i,j=1 aij ei ⊗ ej the Carleman-Fredholm determinant of I + K is deﬁned as det2 (I + K) = det(I + A) exp(−T A),

A.5 Fractional integrals and derivatives

355

where A = (aij )1≤i,j≤n . It can be proved that the mapping K → det2 (I + K) is continuous in L2 (T × T ). As a consequence it can be extended to the whole space L2 (T × T ). If the operator on L2 (T ) associated with the kernel K (denoted again by K) is nuclear, then one can write det2 (I + K) = det(I + K) exp(−T K). A useful formula to compute the Carleman-Fredholm determinant det2 (I + K), where K ∈ L2 (T × T ), is the following: det2 (I + K) = 1 +

∞ γn , n! n=2

(A.12)

where γn =

Tn

ˆ i , tj ))µ(dt1 ) . . . µ(dtn ). det(K(t

ˆ i , tj ) = K(ti , tj ) if i = j and K(t ˆ i , ti ) = 0. Here K(t We refer to [316] and [314] for a more detailed analysis of the properties of this determinant.

A.5 Fractional integrals and derivatives We recall the basic deﬁnitions and properties of the fractional calculus. For a detailed presentation of these notions we refer to [300]. Let a, b ∈ R, a < b. Let f ∈ L1 (a, b) and α > 0. The left and right-sided fractional integrals of f of order α are deﬁned for almost all x ∈ (a, b) by x 1 α−1 Iaα+ f (x) = (x − y) f (y) dy (A.13) Γ (α) a and Ibα− f

1 (x) = Γ (α)

b

α−1

(y − x)

f (y) dy,

(A.14)

x

respectively. Let Iaα+ (Lp ) (resp. Ibα− (Lp )) the image of Lp (a, b) by the operator Iaα+ (resp. Ibα− ). If f ∈ Iaα+ (Lp ) (resp. f ∈ Ibα− (Lp )) and 0 < α < 1 then the left and right-sided fractional derivatives are deﬁned by x f (x) f (x) − f (y) 1 α +α (A.15) Da+ f (x) = α+1 dy , Γ (1 − α) (x − a)α (x − y) a and α Db− f

1 (x) = Γ (1 − α)

f (x) α +α (b − x)

b

x

f (x) − f (y) α+1

(y − x)

dy

(A.16)

356

References

for almost all x ∈ (a, b) (the convergence of the integrals at the singularity y = x holds point-wise for almost all x ∈ (a, b) if p = 1 and moreover in Lp -sense if 1 

1 then p 1

1

where C α− p

Iaα+ (Lp ) ∪ Ibα− (Lp ) ⊂ C α− p (a, b) , ¨ continuous (a, b) denotes the space of α − p1 -Holder

functions of order α −

1 p

in the interval [a, b].

The following inversion formulas hold: α Iaα+ Da+ f =f for all f ∈ Iaα+ (Lp ), and

α α Ia+ f = f Da+

for all f ∈ L1 (a, b). Similar inversion formulas hold for the operators Ibα− α and Db− . The following integration by parts formula holds:

b

α Da+ f (s)g(s)ds =

a

for any f ∈ Iaα+ (Lp ), g ∈ Ibα− (Lq ),

b

α f (s) Db− g (s)ds,

a 1 p

+

1 q

= 1.

(A.17)

References

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Index

absolute continuity Bouleau and Hirsch’s criterion, 97, 163 from duality formula, 92 in dimension one, 98 of the supremum, 109 of transformations of the Wiener measure, 228 on Rm , 90 under Lipschitz coeﬃcients, 127 abstract Wiener space, 81, 225 adapted process, 15, 44 approximate derivative, 94 gradient, 94 partial derivative, 94 approximately diﬀerentiable function, 94 approximation of the identity, 53, 91, 120, 179 arbitrage, 323 Black-Scholes formula, 327

model, 321 Blumenthal zero-one law, 149 Brownian measure on (T, B), 8 Brownian motion, 14 geometric, 327 quadratic variation, 15 Burkholder’s inequality for Hilbert-valued martingales, 352 for two-parameter martingales, 353 Burkholder-Davis-Gundy inequality, 352 Cameron-Martin space, 32 Carleman-Fredholm determinant, 354 Cauchy semigroup, 61 chain rule, 28 for Lipschitz functions, 29 Clark-Ocone formula, 46 closable operator, 26 commutativity relationship for multiplication operators, 65

378

Index

of D and δ, 37 comparison theorem for stochastic parabolic equations, 158, 161 conditional independence, 241 contraction of r indices, 10 symmetric, 11

Fock space, 8 forward Fokker-Planck equation, 129 fractional Brownian motion, 273 transfer principle, 288 fractional derivative, 284, 355 fractional integral, 279, 355

density continuous and bounded, 86 continuously diﬀerentiable, 107, 115 existence of, 87 local criterion for smoothness, 102 positivity, 106 smoothness of, 133 derivative, 325 adjoint operator, 36 covariant, 128 directional, 25 Fr´ ´echet, 32 iterated, 27, 83 of a conditional expectation, 33 of smooth random variables, 25 of the supremum, 109 operator, 25, 82 domain, 27 replicable, 326 trace, 177 Dirichlet boundary conditions, 151 distributions on Gaussian space, 78 divergence integral, 292 divergence operator, 37 Doob’s maximal inequality, 351 duality relation, 37

Gagliardo-Nirenberg inequality, 91 Garsia, Rodemich, and Rumsey’s lemma, 353 Gaussian formula, 351 Gaussian process isonormal, 4, 81 on a Hilbert space H, 4 translation, 94 generalized random variables, 78 Girsanov theorem, 226, 269 anticipating, 234, 240 Girsanov transformation, 218 Greeks, 330 Gronwall’s lemma, 121

elementary function, 8 elementary functions approximation by, 10 factorization property, 258

H¨ older seminorm, 112 H¨ ormander’s condition, 128, 140 H¨ ormander’s theorem, 129 heat equation, 151 Hermite polynomials, 4, 12, 23 generalized, 7 Hilbert space, 4 Hilbert transform, 68 hypercontractivity property, 61 hypoelliptic operator, 129 insider trader, 342 integration-by-parts formula, 25 interest rate, 322 isonormal Gaussian process, 81 Ito’s ˆ formula, 19 for the Skorohod integral, 184 for the Stratonovich integral, 191 kernel square integrable symmetric, 23, 32

Index

summable trace, 177 Kolmogorov’s continuity criterion, 354 Leibnitz rule, 36 Lie algebra, 129, 140 Lie bracket, 128 Lipschitz property for Wiener functionals, 35 Littlewood-Payley inequalities, 83 local property of the Itˆ o integral, 17 of the Skorohod integral, 47 long range dependence, 274 Malliavin matrix, 92 Markov ﬁeld property, 242 germ Markov ﬁeld, 249 process, 242 random ﬁeld, 242 martingale, 18 continuous local, 18, 351 exponential inequality, 352 quadratic variation, 19 martingale measure, 323 mean rate or return, 321 measure L2 (Ω)-valued Gaussian, 8 σ-ﬁnite without atoms, 8 Mehler’s formula, 55 Meyer’s inequalities, 69, 72, 83 multiplier theorem, 64 nonanticipative process, 15 nondegeneracy conditions, 125 Norris’s lemma, 134, 141 option Assian, 326 barrier, 325 digital, 333 European, 328 European call, 325 European put, 325

379

Ornstein-Uhlenbeck generator, 58 process, 56 process parametrized by H, 57 semigroup, 54, 56 Picard’s approximation, 117, 119, 152, 163 Poincar´ ´e’s inequality, 251 portfolio, 322 admissible, 325 self-ﬁnancing, 322 value, 322 price process, 321 discounted, 323 principal value, 70 process self-similar, 273 stationary increments, 274 Radon-Nikodym density, 226, 243, 255, 270 random variable H-continuously diﬀerentiable, 230 H-valued polynomial, 76 polynomial, 7, 25 smooth, 25 ray absolutely continuous, 82 reﬂection principle, 110 reproducing kernel Hilbert space, 253 Riemann sums, 172 semimartingale, 21 weak, 275 Skorohod integral, 40, 47 approximation, 175 continuous version, 181 indeﬁnite, 180 Ito’s ˆ formula, 184 multiple, 53, 83 quadratic variation, 182 substitution formula, 197

380

Index

Sobolev logarithmic inequality, 83 Sobolev space, 77 of Hilbert-valued random variables, 31 of random variables, 27 weighted, 31, 35 stochastic diﬀerential equation anticipating, 208 deﬁnition, 116 existence and uniqueness of solution, 117 in Skorohod sense, 217 in Stratonovich sense, 209 periodic solution, 215 two-point boundary-value problem, 215 weak diﬀerentiability of the solution, 119 with anticipating initial condition, 209 with boundary conditions, 215, 242 stochastic ﬂow, 209 stochastic integral L2 -integral, 177 approximation, 169 forward, 222 indeﬁnite Itˆ oˆ, 18 isometry of the Itˆoˆ integral, 16 Itˆˆo, 16 Itˆ o representation, 22 iterated Itˆˆo, 23 local property of the Itˆ oˆ integral, 17 martingale property, 18 multiple Wiener-Itˆ oˆ, 8, 23, 82 multiplication formula, 11 Skorohod, 40 Stratonovich, 21, 128, 173 Wiener, 8 stochastic integral equations on the plane, 143 stochastic partial diﬀerential equation, 142

elliptic, 250 hyperbolic, 144 parabolic, 151 stochastic process adapted, 15, 44 approximation by step processes, 171 elementary adapted, 15, 44 Markovian, 242 multiparameter, 53 progressively measurable, 16 smooth elementary, 232 square integrable, 15 square integrable adapted, 15 step processes, 171 two-parameter diﬀusion, 143 Stratonovich integral, 21, 128, 173 Ito’s ˆ formula, 191 substitution formula, 199 Stroock’s formula, 35 support of the law, 105, 106, 163 supremum of a continuous process, 109 symmetrization, 9 tensor product, 11 symmetric, 11 triangular function, 248 two-parameter linear stochastic diﬀerential equation, 145 two-parameter Wiener process, 14, 142, 352 utility, 341 volatility, 321 weak topology, 28 white noise, 8 multiparameter, 45 Wiener chaos expansion, 6 derivative of, 32 of order n, 6 orthogonal projection on, 54

Index

Wiener measure, 14 transformation, 225 Wiener sheet, 14 supremum, 111 A∇ j Ak , 128 C, 61 1 , 230 CH C0∞ (Rn ), 25 Cp∞ (Rn ), 25 D, 24 DF , 25 Dh , 27 Dtk1 ,...,tk F , 31 Dt F , 31 F g , 95 H 1 , 32 Hn (x), 4 Im (f ), 9 Jn , 54 L, 58 L2a (T × Ω), 15 L2a (R2+ × Ω), 143 L2a , 44 Qt , 61 S π , 172 Tφ , 63 Tt , 54 [Aj , Ak ], 128 | · |s,p , 77 W 2p p,γ , 80 · L , 60 · k,p,V , 31 · k,p , 27 · H , 4 f p,γ , 112 δ, 36 δ k (u), 53 η(u), 232 γ F , 92 γ i (s), 72 1 Wt , 21, 173 0 ut ◦ dW u ∗ dW W t , 177 T t u dW W , t 16, 40 T t M t , 19

·, ·H , 4 D−∞ , 78 D∞,2 , 35 D∞ , 67 D∞ (V ), 67 Dh,p , 27 Dk,p , 27 Dk,p (V ), 31 D1,p loc , 49 Dh,p loc , 49 Dk,p loc , 49 Ls , 180 L1,2 , 42 2,4 , 193 L− 1,2 , 174 L1,loc 2,4 LC , 191 1,2 Lp+ , 173 1,2 , 173 Lp− Lp1,2 , 173 B0 , 8 Em , 8 F2 , 241 F1 F3

FA , 33 Ft , 15 Hn , 6 HK , 295 P, 24, 25 PH , 76 Pn , 6 Pn0 , 6 S, 25 S0 , 25 Sb , 25 SH , 37 SV , 31 Dom L, 58 Dom δ, 37 T(Du), 177 ap ∇ϕ, 94 ap ϕ (a), 94 ∇ = D+ + D− , 173 ∂α , 100 ∂i , 90

381

382

Index

ρA , 230 |H|, 280 S π , 172 u π (t), 171 f, 9 {W (h), h ∈ H}, 4 f ⊗ g, 11

f ⊗r g, 10 r g, 11 f⊗ 11 f ⊗g, π u (t), 171 det2 (I + K), 355 det2 (I + Du), 232

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